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		<title id="id52896">Symplectic quotients by a nonabelian groupand by its maximal torus</title>
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					<div id="id52856" class="RDFa" property="dct:identifier" content="math.SG/0001002"/>
					<div id="id52865" class="RDFa" property="dct:publisher" content="S. K. Martin &lt;smartin@math.sunysb.edu&gt;"/>
					<div id="id53499" class="RDFa" property="dct:dateSubmitted" content="Sat, 1 Jan 2000 17:48:00 GMT   "/>
					<div id="id53507" class="RDFa" property="dct:title" content="Symplectic quotients by a nonabelian group and by its maximal torus"/>
					<div id="id53517" class="RDFa" property="dct:creator" content="Shaun Martin"/>
					<div id="id53526" class="RDFa" property="dct:subject" content="math.SG"/>
					<div id="id53534" class="RDFa" property="arx:comment" content="18 pages, latex2e. Submitted to Annals of Mathematics Mar 99,  accepted for publication Dec 99Subj-class: Symplectic Geometry"/>
					<h1 id="id53544" class="title document-title">Symplectic quotients by a nonabelian group<br id="id53549" class="break"/>and by its maximal torus</h1>
					<div id="id53555" class="creator author">
						<div id="id53560" class="personname">Shaun Martin<span id="id53564" class="note footnote">
								<sup id="id53568" class="mark">1</sup>
								<div id="id53572" class="note_content">
									<sup id="id53576" class="mark">1</sup>Institute for Advanced Study, Princeton, NJ;
smartin@ias.edu; March, 1999.</div>
							</span>
						</div>
					</div>
					<span id="id53584" class="ERROR undefined">\newdir</span>
					<div class="para" id="p1">
						<p id="id53597" style="" class="p">(*!/-1ex/@^(_(*!/-1ex/@_(<span id="id53604" class="ERROR undefined">\newdir</span>
						</p>
					</div>

					<div class="section" id="Sx1">
						<h2 id="id53617" class="title section-title">Introduction</h2>
						<div class="para" id="Sx1.p1">
							<p id="id53630" style="" class="p">This paper examines the relationship between the symplectic quotient
								<m:math id="id53636" display="inline">
									<m:semantics id="id53640">
										<m:mrow id="id53641">
											<m:mi id="id53646">X</m:mi>
											<m:mo id="id53648">/</m:mo>
											<m:mo id="id53650">/</m:mo>
											<m:mi id="id53652">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53655" encoding="MathML-Content">
											<m:ci id="id53658">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of a Hamiltonian <m:math id="id53662" display="inline">
									<m:semantics id="id53665">
										<m:mi id="id53666">G</m:mi>
										<m:annotation-xml id="id53668" encoding="MathML-Content">
											<m:ci id="id53672">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-manifold <m:math id="id53675" display="inline">
									<m:semantics id="id53678">
										<m:mi id="id53679">X</m:mi>
										<m:annotation-xml id="id53682" encoding="MathML-Content">
											<m:ci id="id53685">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and the associated
symplectic quotient <m:math id="id53689" display="inline">
									<m:semantics id="id53692">
										<m:mrow id="id53693">
											<m:mi id="id53694">X</m:mi>
											<m:mo id="id53696">/</m:mo>
											<m:mo id="id53698">/</m:mo>
											<m:mi id="id53700">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53703" encoding="MathML-Content">
											<m:ci id="id53706">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, where <m:math id="id53710" display="inline">
									<m:semantics id="id53713">
										<m:mrow id="id53714">
											<m:mi id="id53715">T</m:mi>
											<m:mo id="id53717">⊂</m:mo>
											<m:mi id="id53720">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53722" encoding="MathML-Content">
											<m:apply id="id53725">
												<m:ci id="id53726">⊂</m:ci>
												<m:ci id="id53729">T</m:ci>
												<m:ci id="id53731">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a maximal torus,
in the case in which <m:math id="id53735" display="inline">
									<m:semantics id="id53738">
										<m:mrow id="id53739">
											<m:mi id="id53740">X</m:mi>
											<m:mo id="id53742">/</m:mo>
											<m:mo id="id53744">/</m:mo>
											<m:mi id="id53746">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53749" encoding="MathML-Content">
											<m:ci id="id53752">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a compact manifold or
orbifold.</p>
						</div>

						<div class="para" id="Sx1.p2">
							<p id="id53765" style="" class="p">The three main results are: a formula expressing the rational
cohomology ring of <m:math id="id53771" display="inline">
									<m:semantics id="id53774">
										<m:mrow id="id53775">
											<m:mi id="id53776">X</m:mi>
											<m:mo id="id53778">/</m:mo>
											<m:mo id="id53781">/</m:mo>
											<m:mi id="id53783">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53785" encoding="MathML-Content">
											<m:ci id="id53788">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> in terms of the rational cohomology ring of
								<m:math id="id53792" display="inline">
									<m:semantics id="id53795">
										<m:mrow id="id53796">
											<m:mi id="id53797">X</m:mi>
											<m:mo id="id53800">/</m:mo>
											<m:mo id="id53802">/</m:mo>
											<m:mi id="id53804">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53806" encoding="MathML-Content">
											<m:ci id="id53809">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>; an ‘integration’ formula, which expresses cohomology
pairings on <m:math id="id53814" display="inline">
									<m:semantics id="id53817">
										<m:mrow id="id53818">
											<m:mi id="id53819">X</m:mi>
											<m:mo id="id53821">/</m:mo>
											<m:mo id="id53824">/</m:mo>
											<m:mi id="id53826">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53828" encoding="MathML-Content">
											<m:ci id="id53831">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> in terms of cohomology pairings on <m:math id="id53835" display="inline">
									<m:semantics id="id53838">
										<m:mrow id="id53839">
											<m:mi id="id53840">X</m:mi>
											<m:mo id="id53842">/</m:mo>
											<m:mo id="id53844">/</m:mo>
											<m:mi id="id53847">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53849" encoding="MathML-Content">
											<m:ci id="id53852">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>; and an
index formula, which expresses the indices of elliptic operators on
								<m:math id="id53855" display="inline">
									<m:semantics id="id53859">
										<m:mrow id="id53860">
											<m:mi id="id53861">X</m:mi>
											<m:mo id="id53863">/</m:mo>
											<m:mo id="id53865">/</m:mo>
											<m:mi id="id53867">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53869" encoding="MathML-Content">
											<m:ci id="id53873">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> in terms of indices on <m:math id="id53876" display="inline">
									<m:semantics id="id53880">
										<m:mrow id="id53881">
											<m:mi id="id53882">X</m:mi>
											<m:mo id="id53884">/</m:mo>
											<m:mo id="id53886">/</m:mo>
											<m:mi id="id53888">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53890" encoding="MathML-Content">
											<m:ci id="id53894">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="Sx1.p3">
							<p id="id53906" style="" class="p">The results of this paper are complemented by the results in a
companion paper <cite id="id53914" class="cite">[<a id="id53919" href="#bib.bib15" title="" class="ref">15</a>]</cite>, in which different
techniques are used to derive formulæ for cohomology pairings on
symplectic quotients <m:math id="id53932" display="inline">
									<m:semantics id="id53935">
										<m:mrow id="id53936">
											<m:mi id="id53937">X</m:mi>
											<m:mo id="id53939">/</m:mo>
											<m:mo id="id53941">/</m:mo>
											<m:mi id="id53944">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id53946" encoding="MathML-Content">
											<m:ci id="id53949">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, where <m:math id="id53952" display="inline">
									<m:semantics id="id53956">
										<m:mi id="id53957">T</m:mi>
										<m:annotation-xml id="id53959" encoding="MathML-Content">
											<m:ci id="id53962">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a torus, in terms of the
								<m:math id="id53966" display="inline">
									<m:semantics id="id53969">
										<m:mi id="id53970">T</m:mi>
										<m:annotation-xml id="id53972" encoding="MathML-Content">
											<m:ci id="id53976">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-fixed points of <m:math id="id53979" display="inline">
									<m:semantics id="id53983">
										<m:mi id="id53984">X</m:mi>
										<m:annotation-xml id="id53986" encoding="MathML-Content">
											<m:ci id="id53989">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. That paper also gives some applications of
the formulæ proved here.</p>
						</div>

						<div class="para" id="Sx1.p4">
							<p id="id54002" style="" class="p">In order to state the main results of this paper, we introduce some
notation.  The symplectic
quotient <m:math id="id54010" display="inline">
									<m:semantics id="id54013">
										<m:mrow id="id54014">
											<m:mi id="id54015">X</m:mi>
											<m:mo id="id54017">/</m:mo>
											<m:mo id="id54019">/</m:mo>
											<m:mi id="id54021">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id54024" encoding="MathML-Content">
											<m:ci id="id54027">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is defined to be the topological quotient
								<m:math id="id54031" display="inline">
									<m:semantics id="id54034">
										<m:mrow id="id54035">
											<m:mrow id="id54036">
												<m:msubsup id="id54038">
													<m:mi id="id54039">μ</m:mi>
													<m:mi id="id54041">G</m:mi>
													<m:mrow id="id54043">
														<m:mo id="id54044">-</m:mo>
														<m:mn id="id54047">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id54049">⁢</m:mo>
												<m:mfenced id="id54052" open="(" close=")">
													<m:mn id="id52801">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:mo id="id52804">/</m:mo>
											<m:mi id="id52806">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id52808" encoding="MathML-Content">
											<m:apply id="id52811">
												<m:divide id="id52812"/>
												<m:apply id="id52813">
													<m:times id="id52814"/>
													<m:apply id="id52815">
														<m:csymbol id="id52816" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id52821">
															<m:csymbol id="id52822" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id52827">μ</m:ci>
															<m:ci id="id52829">G</m:ci>
														</m:apply>
														<m:apply id="id52831">
															<m:minus id="id52832"/>
															<m:cn id="id52834">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id54263">0</m:cn>
												</m:apply>
												<m:ci id="id54265">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, where <m:math id="id54269" display="inline">
									<m:semantics id="id54272">
										<m:mrow id="id54273">
											<m:msub id="id54274">
												<m:mi id="id54275">μ</m:mi>
												<m:mi id="id54278">G</m:mi>
											</m:msub>
											<m:mo id="id54280">:</m:mo>
											<m:mi id="id54282">X</m:mi>
											<m:mo id="id54284">→</m:mo>
											<m:mo id="id54287">Lie</m:mo>
											<m:mo id="id54289">(</m:mo>
											<m:mi id="id54291">G</m:mi>
											<m:msup id="id54293">
												<m:mo id="id54294">)</m:mo>
												<m:mo id="id54296">*</m:mo>
											</m:msup>
										</m:mrow>
										<m:annotation-xml id="id54298" encoding="MathML-Content">
											<m:apply id="id54302">
												<m:csymbol id="id54303" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id54307">μ</m:ci>
												<m:ci id="id54310">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a moment map for the
								<m:math id="id54313" display="inline">
									<m:semantics id="id54316">
										<m:mi id="id54317">G</m:mi>
										<m:annotation-xml id="id54320" encoding="MathML-Content">
											<m:ci id="id54323">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-action on <m:math id="id54326" display="inline">
									<m:semantics id="id54330">
										<m:mi id="id54331">X</m:mi>
										<m:annotation-xml id="id54333" encoding="MathML-Content">
											<m:ci id="id54336">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. A choice of maximal torus <m:math id="id54339" display="inline">
									<m:semantics id="id54343">
										<m:mrow id="id54344">
											<m:mi id="id54345">T</m:mi>
											<m:mo id="id54347">⊂</m:mo>
											<m:mi id="id54349">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id54352" encoding="MathML-Content">
											<m:apply id="id54355">
												<m:ci id="id54356">⊂</m:ci>
												<m:ci id="id54358">T</m:ci>
												<m:ci id="id54360">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> induces a
natural projection <m:math id="id54364" display="inline">
									<m:semantics id="id54367">
										<m:mrow id="id54368">
											<m:mo id="id54369">Lie</m:mo>
											<m:mo id="id54371">(</m:mo>
											<m:mi id="id54373">G</m:mi>
											<m:msup id="id54376">
												<m:mo id="id54377">)</m:mo>
												<m:mo id="id54379">*</m:mo>
											</m:msup>
											<m:mo id="id54381">↠</m:mo>
											<m:mo id="id54383">Lie</m:mo>
											<m:mo id="id54385">(</m:mo>
											<m:mi id="id54388">T</m:mi>
											<m:msup id="id54390">
												<m:mo id="id54391">)</m:mo>
												<m:mo id="id54393">*</m:mo>
											</m:msup>
										</m:mrow>
										<m:annotation-xml id="id54395" encoding="MathML-Content">
											<m:ci id="id54398">Lie</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and composing with <m:math id="id54402" display="inline">
									<m:semantics id="id54405">
										<m:msub id="id54406">
											<m:mi id="id54407">μ</m:mi>
											<m:mi id="id54410">G</m:mi>
										</m:msub>
										<m:annotation-xml id="id54412" encoding="MathML-Content">
											<m:apply id="id54415">
												<m:csymbol id="id54416" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id54421">μ</m:ci>
												<m:ci id="id54423">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
gives a moment map <m:math id="id54427" display="inline">
									<m:semantics id="id54430">
										<m:mrow id="id54431">
											<m:msub id="id54432">
												<m:mi id="id54433">μ</m:mi>
												<m:mi id="id54436">T</m:mi>
											</m:msub>
											<m:mo id="id54438">:</m:mo>
											<m:mi id="id54440">X</m:mi>
											<m:mo id="id54442">→</m:mo>
											<m:mo id="id54444">Lie</m:mo>
											<m:mo id="id54447">(</m:mo>
											<m:mi id="id54449">T</m:mi>
											<m:msup id="id54451">
												<m:mo id="id54452">)</m:mo>
												<m:mo id="id54454">*</m:mo>
											</m:msup>
										</m:mrow>
										<m:annotation-xml id="id54456" encoding="MathML-Content">
											<m:apply id="id54460">
												<m:csymbol id="id54461" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id54465">μ</m:ci>
												<m:ci id="id54468">T</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for the <m:math id="id54471" display="inline">
									<m:semantics id="id54474">
										<m:mi id="id54476">T</m:mi>
										<m:annotation-xml id="id54478" encoding="MathML-Content">
											<m:ci id="id54481">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-action, with
								<m:math id="id54484" display="inline">
									<m:semantics id="id54488">
										<m:mrow id="id54489">
											<m:mi id="id54490">X</m:mi>
											<m:mo id="id54492">/</m:mo>
											<m:mo id="id54494">/</m:mo>
											<m:mi id="id54496">T</m:mi>
											<m:mo id="id54498">:=</m:mo>
											<m:msubsup id="id54501">
												<m:mi id="id54502">μ</m:mi>
												<m:mi id="id54504">T</m:mi>
												<m:mrow id="id54506">
													<m:mo id="id54507">-</m:mo>
													<m:mn id="id54509">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id54512">(</m:mo>
											<m:mn id="id54514">0</m:mn>
											<m:mo id="id54516">)</m:mo>
											<m:mo id="id54518">/</m:mo>
											<m:mi id="id54520">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id54522" encoding="MathML-Content"/>
									</m:semantics>
								</m:math>. In most of this paper we make some additional
simplifying assumptions: we assume that both <m:math id="id54528" display="inline">
									<m:semantics id="id54531">
										<m:mrow id="id54532">
											<m:msubsup id="id54533">
												<m:mi id="id54534">μ</m:mi>
												<m:mi id="id54536">G</m:mi>
												<m:mrow id="id54539">
													<m:mo id="id54540">-</m:mo>
													<m:mn id="id54542">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id54544">⁢</m:mo>
											<m:mfenced id="id54546" open="(" close=")">
												<m:mn id="id54551">0</m:mn>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id54554" encoding="MathML-Content">
											<m:apply id="id54557">
												<m:times id="id54558"/>
												<m:apply id="id54559">
													<m:csymbol id="id54560" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id54565">
														<m:csymbol id="id54566" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id54570">μ</m:ci>
														<m:ci id="id54573">G</m:ci>
													</m:apply>
													<m:apply id="id54575">
														<m:minus id="id54576"/>
														<m:cn id="id54577">1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn id="id54579">0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math id="id54583" display="inline">
									<m:semantics id="id54586">
										<m:mrow id="id54587">
											<m:msubsup id="id54588">
												<m:mi id="id54589">μ</m:mi>
												<m:mi id="id54592">T</m:mi>
												<m:mrow id="id54594">
													<m:mo id="id54595">-</m:mo>
													<m:mn id="id54597">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id54599">⁢</m:mo>
											<m:mfenced id="id54602" open="(" close=")">
												<m:mn id="id54607">0</m:mn>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id54609" encoding="MathML-Content">
											<m:apply id="id54612">
												<m:times id="id54613"/>
												<m:apply id="id54614">
													<m:csymbol id="id54615" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id54619">
														<m:csymbol id="id54620" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id54625">μ</m:ci>
														<m:ci id="id54627">T</m:ci>
													</m:apply>
													<m:apply id="id54629">
														<m:minus id="id54630"/>
														<m:cn id="id54631">1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn id="id54633">0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are compact manifolds, on which the respective <m:math id="id54637" display="inline">
									<m:semantics id="id54640">
										<m:mi id="id54641">G</m:mi>
										<m:annotation-xml id="id54643" encoding="MathML-Content">
											<m:ci id="id54646">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>- and
								<m:math id="id54649" display="inline">
									<m:semantics id="id54652">
										<m:mi id="id54654">T</m:mi>
										<m:annotation-xml id="id54656" encoding="MathML-Content">
											<m:ci id="id54659">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-actions are free. It follows that <m:math id="id54662" display="inline">
									<m:semantics id="id54665">
										<m:mrow id="id54666">
											<m:mi id="id54667">X</m:mi>
											<m:mo id="id54669">/</m:mo>
											<m:mo id="id54671">/</m:mo>
											<m:mi id="id54674">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id54676" encoding="MathML-Content">
											<m:ci id="id54679">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id54682" display="inline">
									<m:semantics id="id54685">
										<m:mrow id="id54686">
											<m:mi id="id54687">X</m:mi>
											<m:mo id="id54689">/</m:mo>
											<m:mo id="id54691">/</m:mo>
											<m:mi id="id54694">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id54696" encoding="MathML-Content">
											<m:ci id="id54699">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are compact
symplectic manifolds. In section <a id="id54704" href="#S6" title="6 Generalizations ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id54712" style="" class="text tag">6</span>
								</a> we show how to modify
the main results when various of these assumptions are dropped.</p>
						</div>

						<div class="para" id="Sx1.p5">
							<p id="id54727" style="" class="p">For every <span id="id54734" style="" class="text italic">weight</span>
								<m:math id="id54742" display="inline">
									<m:semantics id="id54745">
										<m:mi id="id54746">α</m:mi>
										<m:annotation-xml id="id54748" encoding="MathML-Content">
											<m:ci id="id54752">α</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of <m:math id="id54755" display="inline">
									<m:semantics id="id54759">
										<m:mi id="id54760">T</m:mi>
										<m:annotation-xml id="id54762" encoding="MathML-Content">
											<m:ci id="id54765">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> there is a characteristic class <m:math id="id54769" display="inline">
									<m:semantics id="id54772">
										<m:mrow id="id54773">
											<m:mi id="id54774">e</m:mi>
											<m:mo id="id54776">(</m:mo>
											<m:mi id="id54779">α</m:mi>
											<m:mo id="id54781">)</m:mo>
											<m:mo id="id54783">∈</m:mo>
											<m:msup id="id54786">
												<m:mi id="id54787">H</m:mi>
												<m:mn id="id54789">2</m:mn>
											</m:msup>
											<m:mo id="id54791">(</m:mo>
											<m:mi id="id54793">X</m:mi>
											<m:mo id="id54795">/</m:mo>
											<m:mo id="id54797">/</m:mo>
											<m:mi id="id54799">T</m:mi>
											<m:mo id="id54802">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id54804" encoding="MathML-Content"/>
									</m:semantics>
								</m:math>
naturally associated<span id="id54808" class="note footnote">
									<sup id="id54812" class="mark">2</sup>
									<div id="id54816" class="note_content">
										<sup id="id54819" class="mark">2</sup>A weight of <m:math id="id54825" display="inline">
											<m:semantics id="id54828">
												<m:mi id="id54830">T</m:mi>
												<m:annotation-xml id="id54832" encoding="MathML-Content">
													<m:ci id="id54835">T</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is a homomorphism <m:math id="id54838" display="inline">
											<m:semantics id="id54842">
												<m:mrow id="id54843">
													<m:mi id="id54844">α</m:mi>
													<m:mo id="id54846">:</m:mo>
													<m:mrow id="id54848">
														<m:mi id="id54850">T</m:mi>
														<m:mo id="id54852">→</m:mo>
														<m:msup id="id54854">
															<m:mi id="id54855">S</m:mi>
															<m:mn id="id54857">1</m:mn>
														</m:msup>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id54859" encoding="MathML-Content">
													<m:apply id="id54863">
														<m:ci id="id54864">:</m:ci>
														<m:ci id="id54866">α</m:ci>
														<m:apply id="id54868">
															<m:ci id="id54869">→</m:ci>
															<m:ci id="id54872">T</m:ci>
															<m:apply id="id54874">
																<m:csymbol id="id54875" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id54880">S</m:ci>
																<m:cn id="id54882">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>. Denoting
by <m:math id="id54885" display="inline">
											<m:semantics id="id54889">
												<m:msub id="id54890">
													<m:mi id="id54891" mathvariant="double-struck">C</m:mi>
													<m:mfenced id="id54895" open="(" close=")">
														<m:mi id="id54900">α</m:mi>
													</m:mfenced>
												</m:msub>
												<m:annotation-xml id="id54903" encoding="MathML-Content">
													<m:apply id="id54906">
														<m:csymbol id="id54907" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id54912">C</m:ci>
														<m:ci id="id54914">α</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> the representation space on which <m:math id="id54918" display="inline">
											<m:semantics id="id54921">
												<m:mi id="id54922">T</m:mi>
												<m:annotation-xml id="id54924" encoding="MathML-Content">
													<m:ci id="id54928">T</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> acts via this
homomorphism, we define <m:math id="id54931" display="inline">
											<m:semantics id="id54934">
												<m:mrow id="id54936">
													<m:msub id="id54937">
														<m:mi id="id54938">L</m:mi>
														<m:mi id="id54940">α</m:mi>
													</m:msub>
													<m:mo id="id54942">→</m:mo>
													<m:mi id="id54945">X</m:mi>
													<m:mo id="id54947">/</m:mo>
													<m:mo id="id54949">/</m:mo>
													<m:mi id="id54951">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id54953" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> to be the associated line
bundle, that is, <m:math id="id54958" display="inline">
											<m:semantics id="id54961">
												<m:mrow id="id54962">
													<m:msub id="id54964">
														<m:mi id="id54965">L</m:mi>
														<m:mi id="id54967">α</m:mi>
													</m:msub>
													<m:mo id="id54969">:=</m:mo>
													<m:msubsup id="id54971">
														<m:mi id="id54972">μ</m:mi>
														<m:mi id="id54975">T</m:mi>
														<m:mrow id="id54977">
															<m:mo id="id54978">-</m:mo>
															<m:mn id="id54980">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id54982">(</m:mo>
													<m:mn id="id54984">0</m:mn>
													<m:mo id="id54986">)</m:mo>
													<m:msub id="id54989">
														<m:mo id="id54990">×</m:mo>
														<m:mi id="id54992">T</m:mi>
													</m:msub>
													<m:msub id="id54994">
														<m:mi id="id54995" mathvariant="double-struck">C</m:mi>
														<m:mfenced id="id55000" open="(" close=")">
															<m:mi id="id55005">α</m:mi>
														</m:mfenced>
													</m:msub>
												</m:mrow>
												<m:annotation-xml id="id55007" encoding="MathML-Content">
													<m:apply id="id55010">
														<m:csymbol id="id55012" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id55016">L</m:ci>
														<m:ci id="id55018">α</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and <m:math id="id55022" display="inline">
											<m:semantics id="id55025">
												<m:mrow id="id55026">
													<m:mi id="id55028">e</m:mi>
													<m:mo id="id55030">⁢</m:mo>
													<m:mfenced id="id55032" open="(" close=")">
														<m:mi id="id55037">α</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id55040" encoding="MathML-Content">
													<m:apply id="id55043">
														<m:times id="id55044"/>
														<m:ci id="id55045">e</m:ci>
														<m:ci id="id55047">α</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> to
be the Euler class of <m:math id="id55051" display="inline">
											<m:semantics id="id55054">
												<m:msub id="id55055">
													<m:mi id="id55056">L</m:mi>
													<m:mi id="id55059">α</m:mi>
												</m:msub>
												<m:annotation-xml id="id55061" encoding="MathML-Content">
													<m:apply id="id55064">
														<m:csymbol id="id55065" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id55070">L</m:ci>
														<m:ci id="id55072">α</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</div>
								</span>  to the principal <m:math id="id55077" display="inline">
									<m:semantics id="id55080">
										<m:mi id="id55081">T</m:mi>
										<m:annotation-xml id="id55084" encoding="MathML-Content">
											<m:ci id="id55087">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-bundle
								<m:math id="id55090" display="inline">
									<m:semantics id="id55094">
										<m:mrow id="id55095">
											<m:msubsup id="id55096">
												<m:mi id="id55097">μ</m:mi>
												<m:mi id="id55099">T</m:mi>
												<m:mrow id="id55101">
													<m:mo id="id55102">-</m:mo>
													<m:mn id="id55105">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id55107">(</m:mo>
											<m:mn id="id55109">0</m:mn>
											<m:mo id="id55111">)</m:mo>
											<m:mo id="id55113">→</m:mo>
											<m:mi id="id55116">X</m:mi>
											<m:mo id="id55118">/</m:mo>
											<m:mo id="id55120">/</m:mo>
											<m:mi id="id55122">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id55124" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> (for a precise definition see
section <a id="id55131" href="#S2" title="2 The integration formula ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id55129" style="" class="text tag">2</span>
								</a>).</p>
						</div>

						<div class="theorem theoremA" id="ThmtheoremAx1">
							<h6 id="id55154" class="title theorem-title bold">
								<span id="id55157" class="tag">Theorem A</span> (Cohomology rings).</h6>
							<div class="para" id="ThmtheoremAx1.p1">
								<p id="id55171" style="" class="p">
									<span id="id55176" style="" class="text italic">There is a natural ring isomorphism</span>
								</p>
								<table class="equation" id="Sx1.Ex1">

									<tr valign="baseline" class="equation" id="Sx1.Ex1">
										<td id="id55200" class="eqpad"/>
										<td id="id55204" nowrap="yes" align="center" colspan="1">
											<m:math id="id55212" display="block">
												<m:semantics id="id55215">
													<m:mrow id="id55216">
														<m:msup id="id55217">
															<m:mi id="id55218">H</m:mi>
															<m:mo id="id55220">*</m:mo>
														</m:msup>
														<m:mo id="id55222">(</m:mo>
														<m:mi id="id55224">X</m:mi>
														<m:mo id="id55227">/</m:mo>
														<m:mo id="id55229">/</m:mo>
														<m:mi id="id55231">G</m:mi>
														<m:mo id="id55233">;</m:mo>
														<m:mi id="id55236" mathvariant="double-struck">Q</m:mi>
														<m:mo id="id55240">)</m:mo>
														<m:mo id="id55242">≅</m:mo>
														<m:mfrac id="id55244">
															<m:mrow id="id55246">
																<m:msup id="id55247">
																	<m:mi id="id55248">H</m:mi>
																	<m:mo id="id55250">*</m:mo>
																</m:msup>
																<m:mo id="id55252">(</m:mo>
																<m:mi id="id55254">X</m:mi>
																<m:mo id="id55256">/</m:mo>
																<m:mo id="id55258">/</m:mo>
																<m:mi id="id55260">T</m:mi>
																<m:mo id="id55263">;</m:mo>
																<m:mi id="id55265" mathvariant="double-struck">Q</m:mi>
																<m:msup id="id55269">
																	<m:mo id="id55270">)</m:mo>
																	<m:mi id="id55272">W</m:mi>
																</m:msup>
															</m:mrow>
															<m:mrow id="id55274">
																<m:mo id="id55276">ann</m:mo>
																<m:mo id="id55278">⁡</m:mo>
																<m:mfenced id="id55280" open="(" close=")">
																	<m:mi id="id55285">e</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mfrac>
														<m:mo id="id55287">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id55289" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id55294" class="eqpad"/>
									</tr>

								</table>
								<p id="id55300" style="" class="p">
									<span id="id55305" style="" class="text italic">Here <m:math id="id55312" display="inline">
											<m:semantics id="id55315">
												<m:mi id="id55316">W</m:mi>
												<m:annotation-xml id="id55319" encoding="MathML-Content">
													<m:ci id="id55322">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> denotes the Weyl group of <m:math id="id55326" display="inline">
											<m:semantics id="id55329">
												<m:mi id="id55330">G</m:mi>
												<m:annotation-xml id="id55332" encoding="MathML-Content">
													<m:ci id="id55335">G</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, which acts naturally on
										<m:math id="id55339" display="inline">
											<m:semantics id="id55342">
												<m:mrow id="id55343">
													<m:mi id="id55344">X</m:mi>
													<m:mo id="id55347" mathvariant="normal">/</m:mo>
													<m:mo id="id55351" mathvariant="normal">/</m:mo>
													<m:mi id="id55355">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id55358" encoding="MathML-Content">
													<m:ci id="id55361">X</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>; the class <m:math id="id55364" display="inline">
											<m:semantics id="id55368">
												<m:mrow id="id55369">
													<m:mi id="id55370">e</m:mi>
													<m:mo id="id55372" mathvariant="normal">∈</m:mo>
													<m:msup id="id55377">
														<m:mi id="id55378">H</m:mi>
														<m:mo id="id55380" mathvariant="normal">*</m:mo>
													</m:msup>
													<m:mo id="id55384" mathvariant="normal">(</m:mo>
													<m:mi id="id55389">X</m:mi>
													<m:mo id="id55391" mathvariant="normal">/</m:mo>
													<m:mo id="id55395" mathvariant="normal">/</m:mo>
													<m:mi id="id55400">T</m:mi>
													<m:msup id="id55402">
														<m:mo id="id55403" mathvariant="normal">)</m:mo>
														<m:mi id="id55407">W</m:mi>
													</m:msup>
												</m:mrow>
												<m:annotation-xml id="id55409" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> is defined by
										<m:math id="id55414" display="inline">
											<m:semantics id="id55417">
												<m:mrow id="id55418">
													<m:mi id="id55419">e</m:mi>
													<m:mo id="id55422" mathvariant="normal">:=</m:mo>
													<m:mrow id="id55426">
														<m:msub id="id55427">
															<m:mo id="id55428" mathvariant="normal">∏</m:mo>
															<m:mrow id="id55433">
																<m:mi id="id55434">α</m:mi>
																<m:mo id="id55436" mathvariant="normal">∈</m:mo>
																<m:mi id="id55441" mathvariant="normal">Δ</m:mi>
															</m:mrow>
														</m:msub>
														<m:mi id="id55446">e</m:mi>
														<m:mo id="id55448" mathvariant="italic">⁢</m:mo>
														<m:mfenced id="id55452" open="(" close=")">
															<m:mi id="id55457">α</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id55460" encoding="MathML-Content">
													<m:apply id="id55463">
														<m:ci id="id55464">:=</m:ci>
														<m:ci id="id55466">e</m:ci>
														<m:apply id="id55468">
															<m:apply id="id55470">
																<m:csymbol id="id55471" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id55475">∏</m:ci>
																<m:apply id="id55478">
																	<m:ci id="id55479">∈</m:ci>
																	<m:ci id="id55481">α</m:ci>
																	<m:ci id="id55484">Δ</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id55486">
																<m:times id="id55487"/>
																<m:ci id="id55488">e</m:ci>
																<m:ci id="id55490">α</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> for <m:math id="id55494" display="inline">
											<m:semantics id="id55497">
												<m:mi id="id55498" mathvariant="normal">Δ</m:mi>
												<m:annotation-xml id="id55503" encoding="MathML-Content">
													<m:ci id="id55506">Δ</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> the
roots of <m:math id="id55510" display="inline">
											<m:semantics id="id55513">
												<m:mi id="id55514">G</m:mi>
												<m:annotation-xml id="id55517" encoding="MathML-Content">
													<m:ci id="id55520">G</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and <m:math id="id55523" display="inline">
											<m:semantics id="id55527">
												<m:mrow id="id55528">
													<m:mo id="id55529" mathvariant="normal">ann</m:mo>
													<m:mo id="id55533" mathvariant="normal">(</m:mo>
													<m:mi id="id55538">e</m:mi>
													<m:mo id="id55540" mathvariant="normal">)</m:mo>
													<m:mi id="id55544" mathvariant="normal">⊲</m:mi>
													<m:msup id="id55549">
														<m:mi id="id55550">H</m:mi>
														<m:mo id="id55552" mathvariant="normal">*</m:mo>
													</m:msup>
													<m:mo id="id55556" mathvariant="normal">(</m:mo>
													<m:mi id="id55561">X</m:mi>
													<m:mo id="id55563" mathvariant="normal">/</m:mo>
													<m:mo id="id55567" mathvariant="normal">/</m:mo>
													<m:mi id="id55572">T</m:mi>
													<m:mo id="id55574" mathvariant="normal">;</m:mo>
													<m:mi id="id55578" mathvariant="double-struck">Q</m:mi>
													<m:msup id="id55583">
														<m:mo id="id55584" mathvariant="normal">)</m:mo>
														<m:mi id="id55588">W</m:mi>
													</m:msup>
												</m:mrow>
												<m:annotation-xml id="id55590" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> is the ideal
consisting of all <m:math id="id55595" display="inline">
											<m:semantics id="id55598">
												<m:mi id="id55600">W</m:mi>
												<m:annotation-xml id="id55602" encoding="MathML-Content">
													<m:ci id="id55605">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-invariant elements <m:math id="id55608" display="inline">
											<m:semantics id="id55612">
												<m:mrow id="id55613">
													<m:mi id="id55614">c</m:mi>
													<m:mo id="id55616" mathvariant="normal">∈</m:mo>
													<m:msup id="id55621">
														<m:mi id="id55622">H</m:mi>
														<m:mo id="id55624" mathvariant="normal">*</m:mo>
													</m:msup>
													<m:mo id="id55628" mathvariant="normal">(</m:mo>
													<m:mi id="id55633">X</m:mi>
													<m:mo id="id55635" mathvariant="normal">/</m:mo>
													<m:mo id="id55639" mathvariant="normal">/</m:mo>
													<m:mi id="id55644">T</m:mi>
													<m:mo id="id55646" mathvariant="normal">;</m:mo>
													<m:mi id="id55650" mathvariant="double-struck">Q</m:mi>
													<m:msup id="id55655">
														<m:mo id="id55656" mathvariant="normal">)</m:mo>
														<m:mi id="id55660">W</m:mi>
													</m:msup>
												</m:mrow>
												<m:annotation-xml id="id55662" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> such that
the product <m:math id="id55667" display="inline">
											<m:semantics id="id55670">
												<m:mrow id="id55671">
													<m:mi id="id55672">c</m:mi>
													<m:mo id="id55674" mathvariant="normal">⌣</m:mo>
													<m:mi id="id55679">e</m:mi>
												</m:mrow>
												<m:annotation-xml id="id55681" encoding="MathML-Content">
													<m:apply id="id55685">
														<m:ci id="id55686">⌣</m:ci>
														<m:ci id="id55688">c</m:ci>
														<m:ci id="id55690">e</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> vanishes.</span>
								</p>
							</div>

						</div>

						<div class="para" id="Sx1.p6">
							<p id="id55704" style="" class="p">There is a natural notion of a <span id="id55710" style="" class="text bold">lift</span> of a cohomology class on
								<m:math id="id55718" display="inline">
									<m:semantics id="id55722">
										<m:mrow id="id55723">
											<m:mi id="id55724">X</m:mi>
											<m:mo id="id55726">/</m:mo>
											<m:mo id="id55728">/</m:mo>
											<m:mi id="id55730">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id55732" encoding="MathML-Content">
											<m:ci id="id55736">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> to a class on <m:math id="id55739" display="inline">
									<m:semantics id="id55742">
										<m:mrow id="id55743">
											<m:mi id="id55744">X</m:mi>
											<m:mo id="id55747">/</m:mo>
											<m:mo id="id55749">/</m:mo>
											<m:mi id="id55751">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id55753" encoding="MathML-Content">
											<m:ci id="id55756">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, compatible with the above isomorphism.
The most concrete way of expressing
this<span id="id55760" class="note footnote">
									<sup id="id55764" class="mark">3</sup>
									<div id="id55768" class="note_content">
										<sup id="id55771" class="mark">3</sup>A more natural way of expressing the fact that <m:math id="id55777" display="inline">
											<m:semantics id="id55781">
												<m:mover id="id55782" accent="true">
													<m:mi id="id55786">a</m:mi>
													<m:mo id="id55788">~</m:mo>
												</m:mover>
												<m:annotation-xml id="id55790" encoding="MathML-Content">
													<m:apply id="id55793">
														<m:ci id="id55794">~</m:ci>
														<m:ci id="id55796">a</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
is a lift of <m:math id="id55800" display="inline">
											<m:semantics id="id55803">
												<m:mi id="id55804">a</m:mi>
												<m:annotation-xml id="id55806" encoding="MathML-Content">
													<m:ci id="id55810">a</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> brings in the <m:math id="id55813" display="inline">
											<m:semantics id="id55816">
												<m:mi id="id55818">G</m:mi>
												<m:annotation-xml id="id55820" encoding="MathML-Content">
													<m:ci id="id55823">G</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-equivariant cohomology
										<m:math id="id55827" display="inline">
											<m:semantics id="id55830">
												<m:mrow id="id55831">
													<m:msubsup id="id55832">
														<m:mi id="id55833">H</m:mi>
														<m:mi id="id55835">G</m:mi>
														<m:mo id="id55837">*</m:mo>
													</m:msubsup>
													<m:mo id="id55840">⁢</m:mo>
													<m:mfenced id="id55842" open="(" close=")">
														<m:mi id="id55847">X</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id55849" encoding="MathML-Content">
													<m:apply id="id55852">
														<m:times id="id55854"/>
														<m:apply id="id55855">
															<m:csymbol id="id55856" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id55860">
																<m:csymbol id="id55861" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id55866">H</m:ci>
																<m:ci id="id55868">G</m:ci>
															</m:apply>
															<m:times id="id55870"/>
														</m:apply>
														<m:ci id="id55871">X</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>. There are natural maps from <m:math id="id55875" display="inline">
											<m:semantics id="id55878">
												<m:mrow id="id55879">
													<m:msubsup id="id55880">
														<m:mi id="id55882">H</m:mi>
														<m:mi id="id55884">G</m:mi>
														<m:mo id="id55886">*</m:mo>
													</m:msubsup>
													<m:mo id="id55888">⁢</m:mo>
													<m:mfenced id="id55890" open="(" close=")">
														<m:mi id="id55895">X</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id55898" encoding="MathML-Content">
													<m:apply id="id55901">
														<m:times id="id55902"/>
														<m:apply id="id55903">
															<m:csymbol id="id55904" cd="ambiguous">subscript</m:csymbol>
															<m:apply id="id55909">
																<m:csymbol id="id55910" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id55914">H</m:ci>
																<m:times id="id55917"/>
															</m:apply>
															<m:ci id="id55918">G</m:ci>
														</m:apply>
														<m:ci id="id55920">X</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> to both
										<m:math id="id55923" display="inline">
											<m:semantics id="id55927">
												<m:mrow id="id55928">
													<m:msup id="id55929">
														<m:mi id="id55930">H</m:mi>
														<m:mo id="id55932">*</m:mo>
													</m:msup>
													<m:mo id="id55934">(</m:mo>
													<m:mi id="id55936">X</m:mi>
													<m:mo id="id55938">/</m:mo>
													<m:mo id="id55940">/</m:mo>
													<m:mi id="id55943">G</m:mi>
													<m:mo id="id55945">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id55947" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> and <m:math id="id55952" display="inline">
											<m:semantics id="id55955">
												<m:mrow id="id55956">
													<m:msup id="id55957">
														<m:mi id="id55958">H</m:mi>
														<m:mo id="id55960">*</m:mo>
													</m:msup>
													<m:mo id="id55962">(</m:mo>
													<m:mi id="id55964">X</m:mi>
													<m:mo id="id55967">/</m:mo>
													<m:mo id="id55969">/</m:mo>
													<m:mi id="id55971">T</m:mi>
													<m:mo id="id55973">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id55975" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>, and <m:math id="id55980" display="inline">
											<m:semantics id="id55983">
												<m:mover id="id55984" accent="true">
													<m:mi id="id55988">a</m:mi>
													<m:mo id="id55990">~</m:mo>
												</m:mover>
												<m:annotation-xml id="id55992" encoding="MathML-Content">
													<m:apply id="id55995">
														<m:ci id="id55996">~</m:ci>
														<m:ci id="id55998">a</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is a lift of <m:math id="id56002" display="inline">
											<m:semantics id="id56005">
												<m:mi id="id56006">a</m:mi>
												<m:annotation-xml id="id56008" encoding="MathML-Content">
													<m:ci id="id56012">a</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> if
they are both images of the same class in <m:math id="id56015" display="inline">
											<m:semantics id="id56019">
												<m:mrow id="id56020">
													<m:msubsup id="id56021">
														<m:mi id="id56022">H</m:mi>
														<m:mi id="id56024">G</m:mi>
														<m:mo id="id56026">*</m:mo>
													</m:msubsup>
													<m:mo id="id56028">⁢</m:mo>
													<m:mfenced id="id56031" open="(" close=")">
														<m:mi id="id56036">X</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id56038" encoding="MathML-Content">
													<m:apply id="id56041">
														<m:times id="id56042"/>
														<m:apply id="id56043">
															<m:csymbol id="id56044" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id56049">
																<m:csymbol id="id56050" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id56055">H</m:ci>
																<m:ci id="id56057">G</m:ci>
															</m:apply>
															<m:times id="id56059"/>
														</m:apply>
														<m:ci id="id56060">X</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</div>
								</span>
involves the manifold <m:math id="id56065" display="inline">
									<m:semantics id="id56068">
										<m:mrow id="id56069">
											<m:mi id="id56070">Y</m:mi>
											<m:mo id="id56072">:=</m:mo>
											<m:mrow id="id56074">
												<m:mrow id="id56076">
													<m:msubsup id="id56077">
														<m:mi id="id56078">μ</m:mi>
														<m:mi id="id56080">G</m:mi>
														<m:mrow id="id56082">
															<m:mo id="id56083">-</m:mo>
															<m:mn id="id56085">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id56088">⁢</m:mo>
													<m:mfenced id="id56090" open="(" close=")">
														<m:mn id="id56095">0</m:mn>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id56097">/</m:mo>
												<m:mi id="id56099">T</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id56101" encoding="MathML-Content">
											<m:apply id="id56105">
												<m:ci id="id56106">:=</m:ci>
												<m:ci id="id56108">Y</m:ci>
												<m:apply id="id56110">
													<m:divide id="id56111"/>
													<m:apply id="id56112">
														<m:times id="id56113"/>
														<m:apply id="id56114">
															<m:csymbol id="id56115" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id56120">
																<m:csymbol id="id56121" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id56126">μ</m:ci>
																<m:ci id="id56128">G</m:ci>
															</m:apply>
															<m:apply id="id56130">
																<m:minus id="id56131"/>
																<m:cn id="id56132">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id56135">0</m:cn>
													</m:apply>
													<m:ci id="id56137">T</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. There is an obvious
inclusion <m:math id="id56140" display="inline">
									<m:semantics id="id56144">
										<m:mrow id="id56145">
											<m:mi id="id56146">i</m:mi>
											<m:mo id="id56148">:</m:mo>
											<m:mi id="id56150">Y</m:mi>
											<m:mo id="id56152">↪</m:mo>
											<m:mi id="id56155">X</m:mi>
											<m:mo id="id56157">/</m:mo>
											<m:mo id="id56159">/</m:mo>
											<m:mi id="id56161">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id56163" encoding="MathML-Content">
											<m:ci id="id56166">i</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and projection <m:math id="id56170" display="inline">
									<m:semantics id="id56173">
										<m:mrow id="id56174">
											<m:mi id="id56175">π</m:mi>
											<m:mo id="id56178">:</m:mo>
											<m:mi id="id56180">Y</m:mi>
											<m:mo id="id56182">→</m:mo>
											<m:mi id="id56184">X</m:mi>
											<m:mo id="id56187">/</m:mo>
											<m:mo id="id56189">/</m:mo>
											<m:mi id="id56191">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id56193" encoding="MathML-Content">
											<m:ci id="id56196">π</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and we say
								<m:math id="id56200" display="inline">
									<m:semantics id="id56203">
										<m:mrow id="id56204">
											<m:mover id="id56206" accent="true">
												<m:mi id="id56209">a</m:mi>
												<m:mo id="id56211">~</m:mo>
											</m:mover>
											<m:mo id="id56213">∈</m:mo>
											<m:msup id="id56216">
												<m:mi id="id56217">H</m:mi>
												<m:mo id="id56219">*</m:mo>
											</m:msup>
											<m:mo id="id56221">(</m:mo>
											<m:mi id="id56223">X</m:mi>
											<m:mo id="id56225">/</m:mo>
											<m:mo id="id56227">/</m:mo>
											<m:mi id="id56229">T</m:mi>
											<m:mo id="id56232">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id56234" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> is a <span id="id56238" style="" class="text bold">lift</span> of <m:math id="id56246" display="inline">
									<m:semantics id="id56250">
										<m:mrow id="id56251">
											<m:mi id="id56252">a</m:mi>
											<m:mo id="id56254">∈</m:mo>
											<m:msup id="id56256">
												<m:mi id="id56257">H</m:mi>
												<m:mo id="id56260">*</m:mo>
											</m:msup>
											<m:mo id="id56262">(</m:mo>
											<m:mi id="id56264">X</m:mi>
											<m:mo id="id56266">/</m:mo>
											<m:mo id="id56268">/</m:mo>
											<m:mi id="id56270">G</m:mi>
											<m:mo id="id56272">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id56274" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> if
								<m:math id="id56279" display="inline">
									<m:semantics id="id56282">
										<m:mrow id="id56284">
											<m:mrow id="id56285">
												<m:msup id="id56286">
													<m:mi id="id56287">π</m:mi>
													<m:mo id="id56289">*</m:mo>
												</m:msup>
												<m:mo id="id56291">⁢</m:mo>
												<m:mi id="id56294">a</m:mi>
											</m:mrow>
											<m:mo id="id56296">=</m:mo>
											<m:mrow id="id56298">
												<m:msup id="id56299">
													<m:mi id="id56300">i</m:mi>
													<m:mo id="id56302">*</m:mo>
												</m:msup>
												<m:mo id="id56304">⁢</m:mo>
												<m:mover id="id56307" accent="true">
													<m:mi id="id56310">a</m:mi>
													<m:mo id="id56312">~</m:mo>
												</m:mover>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id56314" encoding="MathML-Content">
											<m:apply id="id56318">
												<m:eq id="id56319"/>
												<m:apply id="id56320">
													<m:times id="id56321"/>
													<m:apply id="id56322">
														<m:csymbol id="id56323" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id56328">π</m:ci>
														<m:times id="id56330"/>
													</m:apply>
													<m:ci id="id56331">a</m:ci>
												</m:apply>
												<m:apply id="id56333">
													<m:times id="id56334"/>
													<m:apply id="id56335">
														<m:csymbol id="id56336" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id56341">i</m:ci>
														<m:times id="id56343"/>
													</m:apply>
													<m:apply id="id56344">
														<m:ci id="id56345">~</m:ci>
														<m:ci id="id56348">a</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="theorem theoremB" id="ThmtheoremBx1">
							<h6 id="id56360" class="title theorem-title bold">
								<span id="id56363" class="tag">Theorem B</span> (Integration formula).</h6>
							<div class="para" id="ThmtheoremBx1.p1">
								<p id="id56376" style="" class="p">
									<span id="id56382" style="" class="text italic">Given a cohomology class  <m:math id="id56388" display="inline">
											<m:semantics id="id56392">
												<m:mrow id="id56393">
													<m:mi id="id56394">a</m:mi>
													<m:mo id="id56396" mathvariant="normal">∈</m:mo>
													<m:msup id="id56401">
														<m:mi id="id56402">H</m:mi>
														<m:mo id="id56404" mathvariant="normal">*</m:mo>
													</m:msup>
													<m:mo id="id56408" mathvariant="normal">(</m:mo>
													<m:mi id="id56413">X</m:mi>
													<m:mo id="id56415" mathvariant="normal">/</m:mo>
													<m:mo id="id56419" mathvariant="normal">/</m:mo>
													<m:mi id="id56424">G</m:mi>
													<m:mo id="id56426" mathvariant="normal">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id56430" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> with lift <m:math id="id56435" display="inline">
											<m:semantics id="id56438">
												<m:mrow id="id56439">
													<m:mover id="id56440" accent="true">
														<m:mi id="id56444">a</m:mi>
														<m:mo id="id56446" mathvariant="normal">~</m:mo>
													</m:mover>
													<m:mo id="id56450" mathvariant="normal">∈</m:mo>
													<m:msup id="id56455">
														<m:mi id="id56456">H</m:mi>
														<m:mo id="id56458" mathvariant="normal">*</m:mo>
													</m:msup>
													<m:mo id="id56462" mathvariant="normal">(</m:mo>
													<m:mi id="id56467">X</m:mi>
													<m:mo id="id56469" mathvariant="normal">/</m:mo>
													<m:mo id="id56473" mathvariant="normal">/</m:mo>
													<m:mi id="id56478">T</m:mi>
													<m:mo id="id56480" mathvariant="normal">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id56484" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>, then</span>
								</p>
								<table class="equation" id="Sx1.Ex2">

									<tr valign="baseline" class="equation" id="Sx1.Ex2">
										<td id="id56506" class="eqpad"/>
										<td id="id56510" nowrap="yes" align="center" colspan="1">
											<m:math id="id56517" display="block">
												<m:semantics id="id56521">
													<m:mrow id="id56522">
														<m:mrow id="id56523">
															<m:mrow id="id56524">
																<m:msub id="id56525">
																	<m:mo id="id56526">∫</m:mo>
																	<m:mrow id="id56528">
																		<m:mi id="id56529">X</m:mi>
																		<m:mo id="id56532">⁣</m:mo>
																		<m:mo id="id56534">/</m:mo>
																		<m:mo id="id56536">⁣</m:mo>
																		<m:mrow id="id56538">
																			<m:none id="id56541"/>
																			<m:mo id="id56542">/</m:mo>
																			<m:mi id="id56544">G</m:mi>
																		</m:mrow>
																	</m:mrow>
																</m:msub>
																<m:mi id="id56546">a</m:mi>
															</m:mrow>
															<m:mo id="id56548">=</m:mo>
															<m:mrow id="id56551">
																<m:mfrac id="id56552">
																	<m:mn id="id56553">1</m:mn>
																	<m:mfenced id="id56555" open="|" close="|">
																		<m:mi id="id56560">W</m:mi>
																	</m:mfenced>
																</m:mfrac>
																<m:mo id="id56562">⁢</m:mo>
																<m:mrow id="id56564">
																	<m:msub id="id56566">
																		<m:mo id="id56567">∫</m:mo>
																		<m:mrow id="id56569">
																			<m:mi id="id56570">X</m:mi>
																			<m:mo id="id56572">⁣</m:mo>
																			<m:mo id="id56575">/</m:mo>
																			<m:mo id="id56577">⁣</m:mo>
																			<m:mrow id="id56579">
																				<m:none id="id56580"/>
																				<m:mo id="id53642">/</m:mo>
																				<m:mi id="id56587">T</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msub>
																	<m:mover id="id56589" accent="true">
																		<m:mi id="id56592">a</m:mi>
																		<m:mo id="id56594">~</m:mo>
																	</m:mover>
																</m:mrow>
															</m:mrow>
															<m:mo id="id56596">⌣</m:mo>
															<m:mi id="id56598">e</m:mi>
														</m:mrow>
														<m:mo id="id56601">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id56603" encoding="MathML-Content">
														<m:apply id="id56606">
															<m:ci id="id56607"/>
															<m:apply id="id56608">
																<m:apply id="id56609">
																	<m:csymbol id="id56610" cd="ambiguous">subscript</m:csymbol>
																	<m:int id="id56615"/>
																	<m:apply id="id56616">
																		<m:list id="id56617"/>
																		<m:ci id="id56618">X</m:ci>
																		<m:divide id="id56620"/>
																		<m:apply id="id56621">
																			<m:divide id="id56622"/>
																			<m:ci id="id56623"/>
																			<m:ci id="id56624">G</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:ci id="id56626">a</m:ci>
															</m:apply>
															<m:eq id="id56629"/>
															<m:apply id="id56630">
																<m:times id="id56631"/>
																<m:apply id="id56632">
																	<m:divide id="id56633"/>
																	<m:cn id="id56634">1</m:cn>
																	<m:apply id="id56636">
																		<m:abs id="id56637"/>
																		<m:ci id="id56638">W</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id56640">
																	<m:apply id="id56641">
																		<m:csymbol id="id56642" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id56647"/>
																		<m:apply id="id56648">
																			<m:list id="id56649"/>
																			<m:ci id="id56650">X</m:ci>
																			<m:divide id="id56652"/>
																			<m:apply id="id56654">
																				<m:divide id="id56655"/>
																				<m:ci id="id56656"/>
																				<m:ci id="id56657">T</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id56659">
																		<m:ci id="id56660">~</m:ci>
																		<m:ci id="id56662">a</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:ci id="id56664">⌣</m:ci>
															<m:ci id="id56667">e</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id56670" class="eqpad"/>
									</tr>

								</table>
								<p id="id56676" style="" class="p">
									<span id="id56681" style="" class="text italic">where <m:math id="id56688" display="inline">
											<m:semantics id="id56691">
												<m:mfenced id="id56692" open="|" close="|">
													<m:mi id="id56698">W</m:mi>
												</m:mfenced>
												<m:annotation-xml id="id56700" encoding="MathML-Content">
													<m:apply id="id56703">
														<m:abs id="id56704"/>
														<m:ci id="id56705">W</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is the order of the Weyl group of <m:math id="id56709" display="inline">
											<m:semantics id="id56712">
												<m:mi id="id56713">G</m:mi>
												<m:annotation-xml id="id56715" encoding="MathML-Content">
													<m:ci id="id56719">G</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and <m:math id="id56722" display="inline">
											<m:semantics id="id56725">
												<m:mi id="id56726">e</m:mi>
												<m:annotation-xml id="id56729" encoding="MathML-Content">
													<m:ci id="id56732">e</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is the
cohomology class defined in Theorem A.</span>
								</p>
							</div>

						</div>

						<div class="para" id="Sx1.p7">
							<p id="id56746" style="" class="p">This formula gives cohomology pairings on <m:math id="id56753" display="inline">
									<m:semantics id="id56756">
										<m:mrow id="id56757">
											<m:mi id="id56758">X</m:mi>
											<m:mo id="id56760">/</m:mo>
											<m:mo id="id56762">/</m:mo>
											<m:mi id="id56764">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id56767" encoding="MathML-Content">
											<m:ci id="id56770">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> because the lift of
a class is compatible with cup product: given classes <m:math id="id56773" display="inline">
									<m:semantics id="id56776">
										<m:mrow id="id56778">
											<m:mi id="id56779">a</m:mi>
											<m:mo id="id56781">,</m:mo>
											<m:mi id="id56783">b</m:mi>
											<m:mo id="id56785">∈</m:mo>
											<m:msup id="id56787">
												<m:mi id="id56788">H</m:mi>
												<m:mo id="id56791">*</m:mo>
											</m:msup>
											<m:mo id="id56793">(</m:mo>
											<m:mi id="id56795">X</m:mi>
											<m:mo id="id56797">/</m:mo>
											<m:mo id="id56799">/</m:mo>
											<m:mi id="id56801">G</m:mi>
											<m:mo id="id56803">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id56806" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> with lifts <m:math id="id56810" display="inline">
									<m:semantics id="id56814">
										<m:mrow id="id56815">
											<m:mover id="id56816" accent="true">
												<m:mi id="id56819">a</m:mi>
												<m:mo id="id56821">~</m:mo>
											</m:mover>
											<m:mo id="id56823">,</m:mo>
											<m:mover id="id56825" accent="true">
												<m:mi id="id56829">b</m:mi>
												<m:mo id="id56831">~</m:mo>
											</m:mover>
											<m:mo id="id56833">∈</m:mo>
											<m:msup id="id56835">
												<m:mi id="id56836">H</m:mi>
												<m:mo id="id56839">*</m:mo>
											</m:msup>
											<m:mo id="id56841">(</m:mo>
											<m:mi id="id56843">X</m:mi>
											<m:mo id="id56845">/</m:mo>
											<m:mo id="id56847">/</m:mo>
											<m:mi id="id56849">T</m:mi>
											<m:mo id="id56851">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id56854" encoding="MathML-Content"/>
									</m:semantics>
								</m:math>, then
								<m:math id="id56858" display="inline">
									<m:semantics id="id56862">
										<m:mrow id="id56863">
											<m:mover id="id56864" accent="true">
												<m:mi id="id56867">a</m:mi>
												<m:mo id="id56869">~</m:mo>
											</m:mover>
											<m:mo id="id56871">⌣</m:mo>
											<m:mover id="id56874" accent="true">
												<m:mi id="id56877">b</m:mi>
												<m:mo id="id56879">~</m:mo>
											</m:mover>
										</m:mrow>
										<m:annotation-xml id="id56881" encoding="MathML-Content">
											<m:apply id="id56885">
												<m:ci id="id56886">⌣</m:ci>
												<m:apply id="id56888">
													<m:ci id="id56889">~</m:ci>
													<m:ci id="id56891">a</m:ci>
												</m:apply>
												<m:apply id="id56893">
													<m:ci id="id56894">~</m:ci>
													<m:ci id="id56897">b</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a lift of <m:math id="id56900" display="inline">
									<m:semantics id="id56903">
										<m:mrow id="id56904">
											<m:mi id="id56906">a</m:mi>
											<m:mo id="id56908">⌣</m:mo>
											<m:mi id="id56910">b</m:mi>
										</m:mrow>
										<m:annotation-xml id="id56912" encoding="MathML-Content">
											<m:apply id="id56916">
												<m:ci id="id56917">⌣</m:ci>
												<m:ci id="id56919">a</m:ci>
												<m:ci id="id56921">b</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="Sx1.p8">
							<p id="id56933" style="" class="p">The symplectic quotient <m:math id="id56940" display="inline">
									<m:semantics id="id56943">
										<m:mrow id="id56944">
											<m:mi id="id56945">X</m:mi>
											<m:mo id="id56947">/</m:mo>
											<m:mo id="id56949">/</m:mo>
											<m:mi id="id56952">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id56954" encoding="MathML-Content">
											<m:ci id="id56957">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a compact symplectic manifold, and
it can be given an almost complex structure
								<m:math id="id56961" display="inline">
									<m:semantics id="id56964">
										<m:mrow id="id56965">
											<m:mi id="id56966">J</m:mi>
											<m:mo id="id56969">:</m:mo>
											<m:mi id="id56971">T</m:mi>
											<m:mo id="id56973">(</m:mo>
											<m:mi id="id56975">X</m:mi>
											<m:mo id="id56977">/</m:mo>
											<m:mo id="id56979">/</m:mo>
											<m:mi id="id56981">G</m:mi>
											<m:mo id="id56984">)</m:mo>
											<m:mo id="id56986">→</m:mo>
											<m:mi id="id56988">T</m:mi>
											<m:mo id="id56990">(</m:mo>
											<m:mi id="id56992">X</m:mi>
											<m:mo id="id56994">/</m:mo>
											<m:mo id="id56997">/</m:mo>
											<m:mi id="id56999">G</m:mi>
											<m:mo id="id57001">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id57003" encoding="MathML-Content">
											<m:ci id="id57006">J</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> which is compatible with its symplectic
structure in a certain sense. Using the same prescription<span id="id57010" class="note footnote">
									<sup id="id57014" class="mark">4</sup>
									<div id="id57018" class="note_content">
										<sup id="id57022" class="mark">4</sup>the operator <m:math id="id57027" display="inline">
											<m:semantics id="id57031">
												<m:mover id="id57032" accent="true">
													<m:mo id="id57035">∂</m:mo>
													<m:mo id="id57037">¯</m:mo>
												</m:mover>
												<m:annotation-xml id="id57040" encoding="MathML-Content">
													<m:apply id="id57043">
														<m:ci id="id57044">¯</m:ci>
														<m:partialdiff id="id57047"/>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is the usual Cauchy-Riemann operator from
										<m:math id="id57049" display="inline">
											<m:semantics id="id57053">
												<m:mfenced id="id57054" open="(" close=")">
													<m:mrow id="id57059">
														<m:mn id="id57060">0</m:mn>
														<m:mo id="id57062">,</m:mo>
														<m:mi id="id57064">i</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:annotation-xml id="id57066" encoding="MathML-Content">
													<m:apply id="id57070">
														<m:interval id="id57071" closure="open"/>
														<m:cn id="id57074">0</m:cn>
														<m:ci id="id57076">i</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-forms to <m:math id="id57080" display="inline">
											<m:semantics id="id57083">
												<m:mfenced id="id57084" open="(" close=")">
													<m:mrow id="id57089">
														<m:mn id="id57090">0</m:mn>
														<m:mo id="id57092">,</m:mo>
														<m:mrow id="id57094">
															<m:mi id="id57095">i</m:mi>
															<m:mo id="id57098">+</m:mo>
															<m:mn id="id57100">1</m:mn>
														</m:mrow>
													</m:mrow>
												</m:mfenced>
												<m:annotation-xml id="id57102" encoding="MathML-Content">
													<m:apply id="id57105">
														<m:interval id="id57106" closure="open"/>
														<m:cn id="id57110">0</m:cn>
														<m:apply id="id57112">
															<m:plus id="id57113"/>
															<m:ci id="id57114">i</m:ci>
															<m:cn id="id57116">1</m:cn>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-forms—this is well-defined on an
almost-complex manifold, although <m:math id="id57120" display="inline">
											<m:semantics id="id57124">
												<m:mrow id="id57125">
													<m:mover id="id57126" accent="true">
														<m:mo id="id57129">∂</m:mo>
														<m:mo id="id57132">¯</m:mo>
													</m:mover>
													<m:mo id="id57134">∘</m:mo>
													<m:mover id="id57136" accent="true">
														<m:mo id="id57140">∂</m:mo>
														<m:mo id="id57142">¯</m:mo>
													</m:mover>
												</m:mrow>
												<m:annotation-xml id="id57145" encoding="MathML-Content">
													<m:apply id="id57148">
														<m:compose id="id57149"/>
														<m:apply id="id57150">
															<m:ci id="id57151">¯</m:ci>
															<m:partialdiff id="id57154"/>
														</m:apply>
														<m:apply id="id57155">
															<m:ci id="id57156">¯</m:ci>
															<m:partialdiff id="id57158"/>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> does not necessarily
vanish—see for example Griffiths and
Harris <cite id="id57162" class="cite">[<a id="id57166" href="#bib.bib7" title="" class="ref">7</a>, p. 80]</cite>; the almost complex structure
combines with the symplectic structure to define a natural metric,
which we use to define <m:math id="id57178" display="inline">
											<m:semantics id="id57181">
												<m:msup id="id57182">
													<m:mover id="id57183" accent="true">
														<m:mo id="id57187">∂</m:mo>
														<m:mo id="id57189">¯</m:mo>
													</m:mover>
													<m:mo id="id57192">*</m:mo>
												</m:msup>
												<m:annotation-xml id="id57194" encoding="MathML-Content">
													<m:apply id="id57197">
														<m:csymbol id="id57198" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id57203">
															<m:ci id="id57204">¯</m:ci>
															<m:partialdiff id="id57206"/>
														</m:apply>
														<m:times id="id57207"/>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.  (The complex structure can
alternatively be viewed as defining a spin<m:math id="id57210" display="inline">
											<m:semantics id="id57214">
												<m:msup id="id57215">
													<m:mi id="id57216"/>
													<m:mi id="id57217">c</m:mi>
												</m:msup>
												<m:annotation-xml id="id57219" encoding="MathML-Content">
													<m:apply id="id57222">
														<m:ci id="id57223">c</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> structure, with <m:math id="id57227" display="inline">
											<m:semantics id="id57230">
												<m:mi id="id57231">D</m:mi>
												<m:annotation-xml id="id57233" encoding="MathML-Content">
													<m:ci id="id57237">D</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
defined as the spin<m:math id="id57240" display="inline">
											<m:semantics id="id57243">
												<m:msup id="id57244">
													<m:mi id="id57246"/>
													<m:mi id="id57247">c</m:mi>
												</m:msup>
												<m:annotation-xml id="id57249" encoding="MathML-Content">
													<m:apply id="id57252">
														<m:ci id="id57253">c</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-Dirac operator, taking even spinors to odd
spinors.)</div>
								</span>
as on a Kähler manifold, we can then define an elliptic operator</p>
							<table class="equation" id="Sx1.Ex3">

								<tr valign="baseline" class="equation" id="Sx1.Ex3">
									<td id="id57276" class="eqpad"/>
									<td id="id57280" nowrap="yes" align="center" colspan="1">
										<m:math id="id57288" display="block">
											<m:semantics id="id57291">
												<m:mrow id="id57292">
													<m:mrow id="id57293">
														<m:mrow id="id57294">
															<m:mi id="id57295">D</m:mi>
															<m:mo id="id57297">:=</m:mo>
															<m:mrow id="id57299">
																<m:mover id="id57300" accent="true">
																	<m:mo id="id57304">∂</m:mo>
																	<m:mo id="id57306">¯</m:mo>
																</m:mover>
																<m:mo id="id57309">+</m:mo>
																<m:msup id="id57311">
																	<m:mover id="id57312" accent="true">
																		<m:mo id="id57315">∂</m:mo>
																		<m:mo id="id57318">¯</m:mo>
																	</m:mover>
																	<m:mo id="id57320">*</m:mo>
																</m:msup>
															</m:mrow>
														</m:mrow>
														<m:mo id="id57322">:</m:mo>
														<m:mrow id="id57324">
															<m:mrow id="id57325">
																<m:msup id="id57326">
																	<m:mi id="id57327">C</m:mi>
																	<m:mi id="id57330" mathvariant="normal">∞</m:mi>
																</m:msup>
																<m:mo id="id57334">⁢</m:mo>
																<m:mfenced id="id57337" open="(" close=")">
																	<m:mrow id="id57342">
																		<m:msup id="id57343">
																			<m:mi id="id57344" mathvariant="normal">Λ</m:mi>
																			<m:mtext id="id57348">even</m:mtext>
																		</m:msup>
																		<m:mo id="id57351">⁢</m:mo>
																		<m:msup id="id57353">
																			<m:mi id="id57354">T</m:mi>
																			<m:mrow id="id57356">
																				<m:mn id="id57358">0</m:mn>
																				<m:mo id="id57360">,</m:mo>
																				<m:mn id="id57362">1</m:mn>
																			</m:mrow>
																		</m:msup>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id57364">→</m:mo>
															<m:mrow id="id57366">
																<m:msup id="id57367">
																	<m:mi id="id57368">C</m:mi>
																	<m:mi id="id57371" mathvariant="normal">∞</m:mi>
																</m:msup>
																<m:mo id="id57375">⁢</m:mo>
																<m:mfenced id="id57378" open="(" close=")">
																	<m:mrow id="id57383">
																		<m:msup id="id57384">
																			<m:mi id="id57385" mathvariant="normal">Λ</m:mi>
																			<m:mtext id="id57390">odd</m:mtext>
																		</m:msup>
																		<m:mo id="id57392">⁢</m:mo>
																		<m:msup id="id57394">
																			<m:mi id="id57395">T</m:mi>
																			<m:mrow id="id57397">
																				<m:mn id="id57398">0</m:mn>
																				<m:mo id="id57400">,</m:mo>
																				<m:mn id="id57403">1</m:mn>
																			</m:mrow>
																		</m:msup>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo id="id57405">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id57407" encoding="MathML-Content">
													<m:apply id="id57410">
														<m:ci id="id57411">:</m:ci>
														<m:apply id="id57413">
															<m:ci id="id57414">:=</m:ci>
															<m:ci id="id57417">D</m:ci>
															<m:apply id="id57419">
																<m:plus id="id57420"/>
																<m:apply id="id57421">
																	<m:ci id="id57422">¯</m:ci>
																	<m:partialdiff id="id57424"/>
																</m:apply>
																<m:apply id="id57425">
																	<m:csymbol id="id57426" cd="ambiguous">superscript</m:csymbol>
																	<m:apply id="id57431">
																		<m:ci id="id57432">¯</m:ci>
																		<m:partialdiff id="id57435"/>
																	</m:apply>
																	<m:times id="id57436"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id57437">
															<m:ci id="id57438">→</m:ci>
															<m:apply id="id57440">
																<m:times id="id57441"/>
																<m:apply id="id57442">
																	<m:csymbol id="id57443" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id57448">C</m:ci>
																	<m:infinity id="id57450"/>
																</m:apply>
																<m:apply id="id57451">
																	<m:times id="id57452"/>
																	<m:apply id="id57453">
																		<m:csymbol id="id57454" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id57459">Λ</m:ci>
																		<m:mtext id="id57462">even</m:mtext>
																	</m:apply>
																	<m:apply id="id57464">
																		<m:csymbol id="id57465" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id57470">T</m:ci>
																		<m:apply id="id57472">
																			<m:list id="id57473"/>
																			<m:cn id="id57474">0</m:cn>
																			<m:cn id="id57476">1</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply id="id57478">
																<m:times id="id57479"/>
																<m:apply id="id57480">
																	<m:csymbol id="id57481" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id57486">C</m:ci>
																	<m:infinity id="id57488"/>
																</m:apply>
																<m:apply id="id57489">
																	<m:times id="id57490"/>
																	<m:apply id="id57491">
																		<m:csymbol id="id57492" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id57497">Λ</m:ci>
																		<m:mtext id="id57500">odd</m:mtext>
																	</m:apply>
																	<m:apply id="id57502">
																		<m:csymbol id="id57503" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id57507">T</m:ci>
																		<m:apply id="id57510">
																			<m:list id="id57511"/>
																			<m:cn id="id57512">0</m:cn>
																			<m:cn id="id57514">1</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td id="id57517" class="eqpad"/>
								</tr>

							</table>
							<p id="id57524" style="" class="p">Furthermore, if
								<m:math id="id57530" display="inline">
									<m:semantics id="id57533">
										<m:mrow id="id57534">
											<m:mi id="id57535">V</m:mi>
											<m:mo id="id57538">→</m:mo>
											<m:mi id="id57540">X</m:mi>
											<m:mo id="id57542">/</m:mo>
											<m:mo id="id57544">/</m:mo>
											<m:mi id="id57546">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id57548" encoding="MathML-Content">
											<m:ci id="id57552">V</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a complex vector bundle, then a choice of Hermitian
connection on <m:math id="id57556" display="inline">
									<m:semantics id="id57559">
										<m:mi id="id57560">V</m:mi>
										<m:annotation-xml id="id57562" encoding="MathML-Content">
											<m:ci id="id57566">V</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> lets us define an elliptic operator
								<m:math id="id57569" display="inline">
									<m:semantics id="id57572">
										<m:mrow id="id57574">
											<m:mrow id="id57575">
												<m:msub id="id57576">
													<m:mi id="id57577">D</m:mi>
													<m:mi id="id57579">V</m:mi>
												</m:msub>
												<m:mo id="id57581">:</m:mo>
												<m:mrow id="id57583">
													<m:mrow id="id57584">
														<m:msup id="id57585">
															<m:mi id="id57586">C</m:mi>
															<m:mi id="id57588" mathvariant="normal">∞</m:mi>
														</m:msup>
														<m:mo id="id57593">⁢</m:mo>
														<m:mfenced id="id57596" open="(" close=")">
															<m:mrow id="id57601">
																<m:mrow id="id57602">
																	<m:mi id="id57603">V</m:mi>
																	<m:mo id="id57605">⊗</m:mo>
																	<m:msup id="id57607">
																		<m:mi id="id57608" mathvariant="normal">Λ</m:mi>
																		<m:mtext id="id57613">even</m:mtext>
																	</m:msup>
																</m:mrow>
																<m:mo id="id57615">⁢</m:mo>
																<m:msup id="id57618">
																	<m:mi id="id57619">T</m:mi>
																	<m:mrow id="id57621">
																		<m:mn id="id57622">0</m:mn>
																		<m:mo id="id57624">,</m:mo>
																		<m:mn id="id57626">1</m:mn>
																	</m:mrow>
																</m:msup>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id57628">→</m:mo>
													<m:mrow id="id57631">
														<m:msup id="id57632">
															<m:mi id="id57633">C</m:mi>
															<m:mi id="id57635" mathvariant="normal">∞</m:mi>
														</m:msup>
														<m:mo id="id57640">⁢</m:mo>
														<m:mfenced id="id57642" open="(" close=")">
															<m:mrow id="id57647">
																<m:mrow id="id57648">
																	<m:mi id="id57649">V</m:mi>
																	<m:mo id="id57652">⊗</m:mo>
																	<m:msup id="id57654">
																		<m:mi id="id57655" mathvariant="normal">Λ</m:mi>
																		<m:mtext id="id57660">odd</m:mtext>
																	</m:msup>
																</m:mrow>
																<m:mo id="id57662">⁢</m:mo>
																<m:msup id="id57664">
																	<m:mi id="id57665">T</m:mi>
																	<m:mrow id="id57667">
																		<m:mn id="id57668">0</m:mn>
																		<m:mo id="id57671">,</m:mo>
																		<m:mn id="id57673">1</m:mn>
																	</m:mrow>
																</m:msup>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
											</m:mrow>
											<m:mo id="id57675">.</m:mo>
										</m:mrow>
										<m:annotation-xml id="id57677" encoding="MathML-Content">
											<m:apply id="id57680">
												<m:ci id="id57681">:</m:ci>
												<m:apply id="id57684">
													<m:csymbol id="id57685" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id57689">D</m:ci>
													<m:ci id="id57691">V</m:ci>
												</m:apply>
												<m:apply id="id57694">
													<m:ci id="id57695">→</m:ci>
													<m:apply id="id57697">
														<m:times id="id57698"/>
														<m:apply id="id57699">
															<m:csymbol id="id57700" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id57705">C</m:ci>
															<m:infinity id="id57707"/>
														</m:apply>
														<m:apply id="id57708">
															<m:times id="id57709"/>
															<m:apply id="id57710">
																<m:ci id="id57711">⊗</m:ci>
																<m:ci id="id57714">V</m:ci>
																<m:apply id="id57716">
																	<m:csymbol id="id57717" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id57722">Λ</m:ci>
																	<m:mtext id="id57724">even</m:mtext>
																</m:apply>
															</m:apply>
															<m:apply id="id57726">
																<m:csymbol id="id57727" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id57732">T</m:ci>
																<m:apply id="id57734">
																	<m:list id="id57735"/>
																	<m:cn id="id57736">0</m:cn>
																	<m:cn id="id57738">1</m:cn>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply id="id57741">
														<m:times id="id57742"/>
														<m:apply id="id57743">
															<m:csymbol id="id57744" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id57748">C</m:ci>
															<m:infinity id="id57751"/>
														</m:apply>
														<m:apply id="id57752">
															<m:times id="id57753"/>
															<m:apply id="id57754">
																<m:ci id="id57755">⊗</m:ci>
																<m:ci id="id57757">V</m:ci>
																<m:apply id="id57759">
																	<m:csymbol id="id57760" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id57765">Λ</m:ci>
																	<m:mtext id="id57768">odd</m:mtext>
																</m:apply>
															</m:apply>
															<m:apply id="id57770">
																<m:csymbol id="id57771" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id57775">T</m:ci>
																<m:apply id="id57778">
																	<m:list id="id57779"/>
																	<m:cn id="id57780">0</m:cn>
																	<m:cn id="id57782">1</m:cn>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
While the operator <m:math id="id57785" display="inline">
									<m:semantics id="id57789">
										<m:msub id="id57790">
											<m:mi id="id57791">D</m:mi>
											<m:mi id="id57793">V</m:mi>
										</m:msub>
										<m:annotation-xml id="id57795" encoding="MathML-Content">
											<m:apply id="id57798">
												<m:csymbol id="id57799" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id57804">D</m:ci>
												<m:ci id="id57806">V</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> depends on the various choices involved, its
index does not, and we have</p>
						</div>

						<div class="theorem theoremC" id="ThmtheoremCx1">
							<h6 id="id57819" class="title theorem-title bold">
								<span id="id57822" class="tag">Theorem C</span> (Index formula).</h6>
							<div class="para" id="ThmtheoremCx1.p1">
								<p id="id57836" style="" class="p">
									<span id="id57841" style="" class="text italic">Suppose <m:math id="id57848" display="inline">
											<m:semantics id="id57851">
												<m:mrow id="id57852">
													<m:mi id="id57853">V</m:mi>
													<m:mo id="id57855" mathvariant="normal">→</m:mo>
													<m:mi id="id57860">X</m:mi>
													<m:mo id="id57862" mathvariant="normal">/</m:mo>
													<m:mo id="id57866" mathvariant="normal">/</m:mo>
													<m:mi id="id57871">G</m:mi>
												</m:mrow>
												<m:annotation-xml id="id57873" encoding="MathML-Content">
													<m:ci id="id57876">V</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is a complex vector bundle, and <m:math id="id57880" display="inline">
											<m:semantics id="id57883">
												<m:mrow id="id57884">
													<m:mover id="id57885" accent="true">
														<m:mi id="id57889">V</m:mi>
														<m:mo id="id57891" mathvariant="normal">~</m:mo>
													</m:mover>
													<m:mo id="id57895" mathvariant="normal">→</m:mo>
													<m:mi id="id57900">X</m:mi>
													<m:mo id="id57902" mathvariant="normal">/</m:mo>
													<m:mo id="id57906" mathvariant="normal">/</m:mo>
													<m:mi id="id57911">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id57913" encoding="MathML-Content">
													<m:apply id="id57916">
														<m:ci id="id57917">~</m:ci>
														<m:ci id="id57920">V</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is a lift of <m:math id="id57923" display="inline">
											<m:semantics id="id57926">
												<m:mi id="id57927">V</m:mi>
												<m:annotation-xml id="id57930" encoding="MathML-Content">
													<m:ci id="id57933">V</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>. Then</span>
								</p>
								<table class="equation" id="Sx1.Ex4">

									<tr valign="baseline" class="equation" id="Sx1.Ex4">
										<td id="id57953" class="eqpad"/>
										<td id="id57957" nowrap="yes" align="center" colspan="1">
											<m:math id="id57965" display="block">
												<m:semantics id="id57968">
													<m:mrow id="id57969">
														<m:mrow id="id57970">
															<m:msup id="id57971">
																<m:mrow id="id57972">
																	<m:mi id="id57973">i</m:mi>
																	<m:mo id="id57976">⁢</m:mo>
																	<m:mi id="id57978">n</m:mi>
																	<m:mo id="id57980">⁢</m:mo>
																	<m:mi id="id57982">d</m:mi>
																	<m:mo id="id57985">⁢</m:mo>
																	<m:mi id="id57987">e</m:mi>
																	<m:mo id="id57989">⁢</m:mo>
																	<m:mi id="id57992">x</m:mi>
																</m:mrow>
																<m:mrow id="id57994">
																	<m:mi id="id57995">X</m:mi>
																	<m:mo id="id57997">⁣</m:mo>
																	<m:mo id="id57999">/</m:mo>
																	<m:mo id="id58001">⁣</m:mo>
																	<m:mrow id="id58004">
																		<m:none id="id58005"/>
																		<m:mo id="id58006">/</m:mo>
																		<m:mi id="id58008">G</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msup>
															<m:mo id="id58010">⁢</m:mo>
															<m:msub id="id58013">
																<m:mi id="id58014">D</m:mi>
																<m:mi id="id58016">V</m:mi>
															</m:msub>
														</m:mrow>
														<m:mo id="id58018">=</m:mo>
														<m:mrow id="id58020">
															<m:mrow id="id58021">
																<m:msup id="id58022">
																	<m:mrow id="id58023">
																		<m:mi id="id58024">i</m:mi>
																		<m:mo id="id58026">⁢</m:mo>
																		<m:mi id="id58029">n</m:mi>
																		<m:mo id="id58031">⁢</m:mo>
																		<m:mi id="id58033">d</m:mi>
																		<m:mo id="id58036">⁢</m:mo>
																		<m:mi id="id58038">e</m:mi>
																		<m:mo id="id58040">⁢</m:mo>
																		<m:mi id="id58042">x</m:mi>
																	</m:mrow>
																	<m:mrow id="id58045">
																		<m:mi id="id58046">X</m:mi>
																		<m:mo id="id58048">⁣</m:mo>
																		<m:mo id="id58050">/</m:mo>
																		<m:mo id="id58052">⁣</m:mo>
																		<m:mrow id="id58055">
																			<m:none id="id58056"/>
																			<m:mo id="id58057">/</m:mo>
																			<m:mi id="id58059">T</m:mi>
																		</m:mrow>
																	</m:mrow>
																</m:msup>
																<m:mo id="id58061">⁢</m:mo>
																<m:msub id="id58064">
																	<m:mi id="id58065">D</m:mi>
																	<m:mrow id="id58067">
																		<m:mrow id="id58068">
																			<m:mover id="id58069" accent="true">
																				<m:mi id="id58072">V</m:mi>
																				<m:mo id="id58074">~</m:mo>
																			</m:mover>
																			<m:mo id="id58076">⊗</m:mo>
																			<m:msup id="id58079">
																				<m:mi id="id58080" mathvariant="normal">Λ</m:mi>
																				<m:mtext id="id58085" mathvariant="italic">even</m:mtext>
																			</m:msup>
																		</m:mrow>
																		<m:mo id="id58089">⁢</m:mo>
																		<m:mi id="id58092">E</m:mi>
																	</m:mrow>
																</m:msub>
															</m:mrow>
															<m:mo id="id58094">-</m:mo>
															<m:mrow id="id58096">
																<m:msup id="id58097">
																	<m:mrow id="id58098">
																		<m:mi id="id58099">i</m:mi>
																		<m:mo id="id58101">⁢</m:mo>
																		<m:mi id="id58104">n</m:mi>
																		<m:mo id="id58106">⁢</m:mo>
																		<m:mi id="id58108">d</m:mi>
																		<m:mo id="id58110">⁢</m:mo>
																		<m:mi id="id58113">e</m:mi>
																		<m:mo id="id58115">⁢</m:mo>
																		<m:mi id="id58117">x</m:mi>
																	</m:mrow>
																	<m:mrow id="id58119">
																		<m:mi id="id58120">X</m:mi>
																		<m:mo id="id58123">⁣</m:mo>
																		<m:mo id="id58125">/</m:mo>
																		<m:mo id="id58127">⁣</m:mo>
																		<m:mrow id="id58130">
																			<m:none id="id58131"/>
																			<m:mo id="id58132">/</m:mo>
																			<m:mi id="id58134">T</m:mi>
																		</m:mrow>
																	</m:mrow>
																</m:msup>
																<m:mo id="id58136">⁢</m:mo>
																<m:msub id="id58138">
																	<m:mi id="id58139">D</m:mi>
																	<m:mrow id="id58142">
																		<m:mrow id="id58143">
																			<m:mover id="id58144" accent="true">
																				<m:mi id="id58147">V</m:mi>
																				<m:mo id="id58149">~</m:mo>
																			</m:mover>
																			<m:mo id="id58151">⊗</m:mo>
																			<m:msup id="id58154">
																				<m:mi id="id58155" mathvariant="normal">Λ</m:mi>
																				<m:mtext id="id58159" mathvariant="italic">odd</m:mtext>
																			</m:msup>
																		</m:mrow>
																		<m:mo id="id58164">⁢</m:mo>
																		<m:mi id="id58166">E</m:mi>
																	</m:mrow>
																</m:msub>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id58168" encoding="MathML-Content">
														<m:apply id="id58172">
															<m:eq id="id58173"/>
															<m:apply id="id58174">
																<m:times id="id58175"/>
																<m:apply id="id58176">
																	<m:csymbol id="id58177" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id58182"/>
																	<m:apply id="id58183">
																		<m:list id="id58184"/>
																		<m:ci id="id58185">X</m:ci>
																		<m:divide id="id58187"/>
																		<m:apply id="id58188">
																			<m:divide id="id58189"/>
																			<m:ci id="id58190"/>
																			<m:ci id="id58191">G</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id58193">
																	<m:csymbol id="id58194" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id58199">D</m:ci>
																	<m:ci id="id58201">V</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id58203">
																<m:minus id="id58204"/>
																<m:apply id="id58206">
																	<m:times id="id58207"/>
																	<m:apply id="id58208">
																		<m:csymbol id="id58209" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id58213"/>
																		<m:apply id="id58214">
																			<m:list id="id58216"/>
																			<m:ci id="id58217">X</m:ci>
																			<m:divide id="id58219"/>
																			<m:apply id="id58220">
																				<m:divide id="id58221"/>
																				<m:ci id="id58222"/>
																				<m:ci id="id58223">T</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id58225">
																		<m:csymbol id="id58226" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id58231">D</m:ci>
																		<m:apply id="id58233">
																			<m:times id="id58234"/>
																			<m:apply id="id58235">
																				<m:ci id="id58236">⊗</m:ci>
																				<m:apply id="id58239">
																					<m:ci id="id58240">~</m:ci>
																					<m:ci id="id58242">V</m:ci>
																				</m:apply>
																				<m:apply id="id58244">
																					<m:csymbol id="id58245" cd="ambiguous">superscript</m:csymbol>
																					<m:ci id="id58250">Λ</m:ci>
																					<m:mtext id="id58252">even</m:mtext>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id58254">E</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id58257">
																	<m:times id="id58258"/>
																	<m:apply id="id58259">
																		<m:csymbol id="id58260" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id58264"/>
																		<m:apply id="id58266">
																			<m:list id="id58267"/>
																			<m:ci id="id58268">X</m:ci>
																			<m:divide id="id58270"/>
																			<m:apply id="id58271">
																				<m:divide id="id58272"/>
																				<m:ci id="id58273"/>
																				<m:ci id="id58274">T</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id58276">
																		<m:csymbol id="id58277" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id58282">D</m:ci>
																		<m:apply id="id58284">
																			<m:times id="id58285"/>
																			<m:apply id="id58286">
																				<m:ci id="id58287">⊗</m:ci>
																				<m:apply id="id58290">
																					<m:ci id="id58291">~</m:ci>
																					<m:ci id="id58293">V</m:ci>
																				</m:apply>
																				<m:apply id="id58295">
																					<m:csymbol id="id58296" cd="ambiguous">superscript</m:csymbol>
																					<m:ci id="id58301">Λ</m:ci>
																					<m:mtext id="id58303">odd</m:mtext>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id58305">E</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id58308" class="eqpad"/>
									</tr>

								</table>
								<p id="id58315" style="" class="p">
									<span id="id58320" style="" class="text italic">Here we can take <m:math id="id58327" display="inline">
											<m:semantics id="id58330">
												<m:mrow id="id58331">
													<m:mi id="id58332">E</m:mi>
													<m:mo id="id58334" mathvariant="normal">→</m:mo>
													<m:mi id="id58339">X</m:mi>
													<m:mo id="id58341" mathvariant="normal">/</m:mo>
													<m:mo id="id58346" mathvariant="normal">/</m:mo>
													<m:mi id="id58350">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id58352" encoding="MathML-Content">
													<m:ci id="id58355">E</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> to equal <m:math id="id58359" display="inline">
											<m:semantics id="id58362">
												<m:mrow id="id58363">
													<m:msub id="id58364">
														<m:mo id="id58365" mathvariant="normal">⊕</m:mo>
														<m:mrow id="id58370">
															<m:mi id="id58371">α</m:mi>
															<m:mo id="id58374" mathvariant="normal">∈</m:mo>
															<m:msup id="id58378">
																<m:mi id="id58379" mathvariant="normal">Δ</m:mi>
																<m:mo id="id58384" mathvariant="normal">+</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:msub id="id58388">
														<m:mi id="id58389">L</m:mi>
														<m:mi id="id58392">α</m:mi>
													</m:msub>
												</m:mrow>
												<m:annotation-xml id="id58394" encoding="MathML-Content">
													<m:apply id="id58397">
														<m:apply id="id58398">
															<m:csymbol id="id58399" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id58404">⊕</m:ci>
															<m:apply id="id58406">
																<m:ci id="id58408">∈</m:ci>
																<m:ci id="id58410">α</m:ci>
																<m:apply id="id58412">
																	<m:csymbol id="id58413" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id58418">Δ</m:ci>
																	<m:plus id="id58420"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id58422">
															<m:csymbol id="id58423" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id58427">L</m:ci>
															<m:ci id="id58429">α</m:ci>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> for
any choice <m:math id="id58433" display="inline">
											<m:semantics id="id58436">
												<m:msup id="id58438">
													<m:mi id="id58439" mathvariant="normal">Δ</m:mi>
													<m:mo id="id58443" mathvariant="normal">+</m:mo>
												</m:msup>
												<m:annotation-xml id="id58448" encoding="MathML-Content">
													<m:apply id="id58451">
														<m:csymbol id="id58452" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id58457">Δ</m:ci>
														<m:plus id="id58459"/>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> of positive roots of <m:math id="id58462" display="inline">
											<m:semantics id="id58465">
												<m:mi id="id58466">G</m:mi>
												<m:annotation-xml id="id58468" encoding="MathML-Content">
													<m:ci id="id58471">G</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, where <m:math id="id58475" display="inline">
											<m:semantics id="id58478">
												<m:mrow id="id58479">
													<m:msub id="id58480">
														<m:mi id="id58481">L</m:mi>
														<m:mi id="id58484">α</m:mi>
													</m:msub>
													<m:mo id="id58486" mathvariant="normal">→</m:mo>
													<m:mi id="id58491">X</m:mi>
													<m:mo id="id58493" mathvariant="normal">/</m:mo>
													<m:mo id="id58497" mathvariant="normal">/</m:mo>
													<m:mi id="id58502">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id58504" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
denotes the complex line bundle naturally associated to the weight <m:math id="id58509" display="inline">
											<m:semantics id="id58512">
												<m:mi id="id58513">α</m:mi>
												<m:annotation-xml id="id58516" encoding="MathML-Content">
													<m:ci id="id58519">α</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</span>
								</p>
							</div>

						</div>

						<div class="para" id="Sx1.p9">
							<p id="id58532" style="" class="p">This formula is simpler than one would get by applying the
Atiyah-Singer index theorem to the integration formula—its proof
runs along similar lines, but brings in a result of Borel and
Hirzebruch on the Todd genus of <m:math id="id58542" display="inline">
									<m:semantics id="id58546">
										<m:mrow id="id58547">
											<m:mi id="id58548">G</m:mi>
											<m:mo id="id58550">/</m:mo>
											<m:mi id="id58552">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id58554" encoding="MathML-Content">
											<m:apply id="id58557">
												<m:divide id="id58558"/>
												<m:ci id="id58560">G</m:ci>
												<m:ci id="id58562">T</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="Sx1.p10">
							<p id="id58574" style="" class="p">The layout of this paper is as follows. Section <a id="id58583" href="#S1" title="1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58581" style="" class="text tag">1</span>
								</a> contains
the main topological result, giving a detailed description of the
topological relationship between <m:math id="id58598" display="inline">
									<m:semantics id="id58601">
										<m:mrow id="id58602">
											<m:mi id="id58603">X</m:mi>
											<m:mo id="id58606">/</m:mo>
											<m:mo id="id58608">/</m:mo>
											<m:mi id="id58610">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id58612" encoding="MathML-Content">
											<m:ci id="id58615">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and
								<m:math id="id58618" display="inline">
									<m:semantics id="id58622">
										<m:mrow id="id58623">
											<m:mi id="id58624">X</m:mi>
											<m:mo id="id58626">/</m:mo>
											<m:mo id="id58628">/</m:mo>
											<m:mi id="id58630">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id58632" encoding="MathML-Content">
											<m:ci id="id58636">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Section <a id="id58641" href="#S2" title="2 The integration formula ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58639" style="" class="text tag">2</span>
								</a> uses this result, together with some
cohomological facts, to prove theorem B, the integration formula.  In
section <a id="id58659" href="#S3" title="3 The relationship between the cohomology rings ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58657" style="" class="text tag">3</span>
								</a> the integration formula is combined with
Poincaré duality to prove theorem A, and in section <a id="id58676" href="#S4" title="4 The index formula ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58684" style="" class="text tag">4</span>
								</a> the
index formula is proved. Section <a id="id58693" href="#S5" title="5 Characteristic numbers ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58692" style="" class="text tag">5</span>
								</a> is a very short section
describing some formulæ for characteristic numbers of <m:math id="id58709" display="inline">
									<m:semantics id="id58712">
										<m:mrow id="id58713">
											<m:mi id="id58714">X</m:mi>
											<m:mo id="id58716">/</m:mo>
											<m:mo id="id58718">/</m:mo>
											<m:mi id="id58721">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id58723" encoding="MathML-Content">
											<m:ci id="id58726">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, such
as the Euler characteristic and the signature. In
section <a id="id58731" href="#S6" title="6 Generalizations ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58739" style="" class="text tag">6</span>
								</a> various generalizations of the main results are
described, including straightforward generalizations to the case when
the two symplectic quotients are compact orbifolds, to the case in
which <m:math id="id58747" display="inline">
									<m:semantics id="id58751">
										<m:mrow id="id58752">
											<m:mi id="id58753">X</m:mi>
											<m:mo id="id58755">/</m:mo>
											<m:mo id="id58757">/</m:mo>
											<m:mi id="id58759">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id58761" encoding="MathML-Content">
											<m:ci id="id58765">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> may have singularities or be noncompact, and finally, in
a different direction, a generalization in which <m:math id="id58769" display="inline">
									<m:semantics id="id58772">
										<m:mi id="id58773">T</m:mi>
										<m:annotation-xml id="id58776" encoding="MathML-Content">
											<m:ci id="id58779">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is replaced by a
subgroup of full rank. Finally, in
section <a id="id58785" href="#S7" title="7 Example: The complex Grassmannian ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id58783" style="" class="text tag">7</span>
								</a>, the results of this paper are applied
to the explicit example of the Grassmannian of <m:math id="id58800" display="inline">
									<m:semantics id="id58803">
										<m:mi id="id58804">k</m:mi>
										<m:annotation-xml id="id58806" encoding="MathML-Content">
											<m:ci id="id58810">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-dimensional planes
in <m:math id="id58813" display="inline">
									<m:semantics id="id58816">
										<m:msup id="id58817">
											<m:mi id="id58818" mathvariant="double-struck">C</m:mi>
											<m:mi id="id58823">n</m:mi>
										</m:msup>
										<m:annotation-xml id="id58825" encoding="MathML-Content">
											<m:apply id="id58828">
												<m:csymbol id="id58829" cd="ambiguous">superscript</m:csymbol>
												<m:ci id="id58834">C</m:ci>
												<m:ci id="id58836">n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which arises as a symplectic quotient by the group <m:math id="id58839" display="inline">
									<m:semantics id="id58843">
										<m:mrow id="id58844">
											<m:mi id="id58845">U</m:mi>
											<m:mo id="id58847">⁢</m:mo>
											<m:mfenced id="id58849" open="(" close=")">
												<m:mi id="id58854">k</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id58857" encoding="MathML-Content">
											<m:apply id="id58860">
												<m:times id="id58861"/>
												<m:ci id="id58862">U</m:ci>
												<m:ci id="id58864">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The associated symplectic quotient by the maximal torus is the
								<m:math id="id58868" display="inline">
									<m:semantics id="id58871">
										<m:mi id="id58872">k</m:mi>
										<m:annotation-xml id="id58875" encoding="MathML-Content">
											<m:ci id="id58878">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-fold product <m:math id="id58881" display="inline">
									<m:semantics id="id58885">
										<m:msup id="id58886">
											<m:mfenced id="id58887" open="(" close=")">
												<m:mrow id="id58892">
													<m:mi id="id58893" mathvariant="double-struck">C</m:mi>
													<m:mo id="id58897">⁢</m:mo>
													<m:msup id="id58900">
														<m:mi id="id58901" mathvariant="double-struck">P</m:mi>
														<m:mrow id="id58906">
															<m:mi id="id58907">n</m:mi>
															<m:mo id="id58909">-</m:mo>
															<m:mn id="id58911">1</m:mn>
														</m:mrow>
													</m:msup>
												</m:mrow>
											</m:mfenced>
											<m:mi id="id58913">k</m:mi>
										</m:msup>
										<m:annotation-xml id="id58915" encoding="MathML-Content">
											<m:apply id="id58919">
												<m:csymbol id="id58920" cd="ambiguous">superscript</m:csymbol>
												<m:apply id="id58924">
													<m:times id="id58925"/>
													<m:ci id="id58926">C</m:ci>
													<m:apply id="id58929">
														<m:csymbol id="id58930" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id58934">P</m:ci>
														<m:apply id="id58936">
															<m:minus id="id58938"/>
															<m:ci id="id58939">n</m:ci>
															<m:cn id="id58941">1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:ci id="id58943">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. One result is a
presentation of the cohomology of the Grassmannian which is
different from the usual one.</p>
						</div>

						<div class="subsection" id="Sx1.SSx1">
							<h3 id="id58956" class="title subsection-title">Relation to other results</h3>
							<div class="para" id="Sx1.SSx1.p1">
								<p id="id58968" style="" class="p">The results of this paper, together with the companion
paper <cite id="id58976" class="cite">[<a id="id58980" href="#bib.bib15" title="" class="ref">15</a>]</cite>, have been applied by Jeffrey and
Kirwan <cite id="id58992" class="cite">[<a id="id58996" href="#bib.bib11" title="" class="ref">11</a>]</cite>  to prove certain formulæ
for cohomology pairings on moduli spaces of stable holomorphic bundles
over a Riemann surface. These formulæ were first derived by
Witten <cite id="id59009" class="cite">[<a id="id59014" href="#bib.bib21" title="" class="ref">21</a>]</cite> using physical arguments. Indeed the main
motivation for the results of this paper and its companion was to find
a purely topological proof of Witten’s formulæ.</p>
							</div>

							<div class="para" id="Sx1.SSx1.p2">
								<p id="id59036" style="" class="p">This work was carried out in 1994, in Oxford, and at the Isaac Newton
Institute in Cambridge. After completing this work, I was made aware
of some related results of Ellingsrud and Str∘mme <cite id="id59046" class="cite">[<a id="id59050" href="#bib.bib5" title="" class="ref">5</a>]</cite>. The present paper intersects with theirs
in the case that <m:math id="id59061" display="inline">
										<m:semantics id="id59064">
											<m:mi id="id59066">X</m:mi>
											<m:annotation-xml id="id59068" encoding="MathML-Content">
												<m:ci id="id59071">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a complex vector space; in that case they have
a result closely related to theorem A. (Their general setting
is the ‘Geometric invariant theory’
quotient <cite id="id59077" class="cite">[<a id="id59082" href="#bib.bib19" title="" class="ref">19</a>]</cite> of a vector space over an
arbitrary field, for which they calculate the Chow ring; their
techniques are completely different from those used here.)</p>
							</div>

						</div>

						<div class="subsection" id="Sx1.SSx2">
							<h3 id="id59103" class="title subsection-title">Acknowledgements</h3>
							<div class="para" id="Sx1.SSx2.p1">
								<p id="id59116" style="" class="p">It is a pleasure to thank Simon Donaldson, Robert Purves,
John Rawnsley and Dietmar
Salamon for illuminating discussions, and Lisa Jeffrey and Frances
Kirwan for their interest and encouragement while these results were
being developed.
								</p>
							</div>

						</div>

						<div class="subsection" id="Sx1.SSx3">
							<h3 id="id59134" class="title subsection-title">Notation</h3>
							<div class="para" id="Sx1.SSx3.p1">
								<p id="id59146" style="" class="p">Fixed throughout this paper are the following:</p>

								<ul id="id59154" class="itemize">

									<li class="item nobullet" id="I1.i1">
										<span id="id59165" class="tag">
											<m:math id="id59168" display="inline">
												<m:semantics id="id59171">
													<m:mi id="id59172">X</m:mi>
													<m:annotation-xml id="id59175" encoding="MathML-Content">
														<m:ci id="id59178">X</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i1.p1">
											<p id="id59188" style="" class="p">is a fixed smooth symplectic manifold (with symplectic form <m:math id="id59195" display="inline">
													<m:semantics id="id59198">
														<m:mi id="id59199">ω</m:mi>
														<m:annotation-xml id="id59201" encoding="MathML-Content">
															<m:ci id="id59205">ω</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>);</p>
										</div>

									</li>

									<li class="item nobullet" id="I1.i2">
										<span id="id59218" class="tag">
											<m:math id="id59222" display="inline">
												<m:semantics id="id59225">
													<m:mrow id="id59226">
														<m:mi id="id59227">G</m:mi>
														<m:mo id="id59229">⊃</m:mo>
														<m:mi id="id59232">T</m:mi>
													</m:mrow>
													<m:annotation-xml id="id59234" encoding="MathML-Content">
														<m:apply id="id59237">
															<m:ci id="id59238">⊃</m:ci>
															<m:ci id="id59240">G</m:ci>
															<m:ci id="id59243">T</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i2.p1">
											<p id="id59252" style="" class="p">is a connected compact Lie group and a fixed
maximal torus, both acting on <m:math id="id59259" display="inline">
													<m:semantics id="id59263">
														<m:mi id="id59264">X</m:mi>
														<m:annotation-xml id="id59266" encoding="MathML-Content">
															<m:ci id="id59269">X</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, preserving <m:math id="id59273" display="inline">
													<m:semantics id="id59276">
														<m:mi id="id59277">ω</m:mi>
														<m:annotation-xml id="id59280" encoding="MathML-Content">
															<m:ci id="id59283">ω</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>;</p>
										</div>

									</li>

									<li class="item nobullet" id="I1.i3">
										<span id="id59296" class="tag">
											<m:math id="id59299" display="inline">
												<m:semantics id="id59303">
													<m:mrow id="id59304">
														<m:mi id="id59305" mathvariant="fraktur">g</m:mi>
														<m:mo id="id59310">⊃</m:mo>
														<m:mi id="id59312" mathvariant="fraktur">t</m:mi>
													</m:mrow>
													<m:annotation-xml id="id59316" encoding="MathML-Content">
														<m:apply id="id59320">
															<m:ci id="id59321">⊃</m:ci>
															<m:ci id="id59323">g</m:ci>
															<m:ci id="id59325">t</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i3.p1">
											<p id="id59335" style="" class="p">are their Lie algebras;</p>
										</div>

									</li>

									<li class="item nobullet" id="I1.i4">
										<span id="id59351" class="tag">
											<m:math id="id59354" display="inline">
												<m:semantics id="id59358">
													<m:mrow id="id59359">
														<m:msub id="id59360">
															<m:mi id="id59361">μ</m:mi>
															<m:mi id="id59363">G</m:mi>
														</m:msub>
														<m:mo id="id59366">:</m:mo>
														<m:mrow id="id59368">
															<m:mi id="id59369">X</m:mi>
															<m:mo id="id59371">→</m:mo>
															<m:msup id="id59373">
																<m:mi id="id59374" mathvariant="fraktur">g</m:mi>
																<m:mo id="id59379">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id59381" encoding="MathML-Content">
														<m:apply id="id59384">
															<m:ci id="id59385">:</m:ci>
															<m:apply id="id59387">
																<m:csymbol id="id59388" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id59393">μ</m:ci>
																<m:ci id="id59396">G</m:ci>
															</m:apply>
															<m:apply id="id59398">
																<m:ci id="id59399">→</m:ci>
																<m:ci id="id59401">X</m:ci>
																<m:apply id="id59403">
																	<m:csymbol id="id59404" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id59409">g</m:ci>
																	<m:times id="id59411"/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i4.p1">
											<p id="id59420" style="" class="p">is a moment map for the <m:math id="id59426" display="inline">
													<m:semantics id="id59430">
														<m:mi id="id59431">G</m:mi>
														<m:annotation-xml id="id59433" encoding="MathML-Content">
															<m:ci id="id59436">G</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> action on
												<m:math id="id59440" display="inline">
													<m:semantics id="id59443">
														<m:mi id="id59444">X</m:mi>
														<m:annotation-xml id="id59446" encoding="MathML-Content">
															<m:ci id="id59450">X</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, which we assume throughout to be proper, and to have <m:math id="id59453" display="inline">
													<m:semantics id="id59457">
														<m:mn id="id59458">0</m:mn>
														<m:annotation-xml id="id59460" encoding="MathML-Content">
															<m:cn id="id59463">0</m:cn>
														</m:annotation-xml>
													</m:semantics>
												</m:math> as a
regular value (our sign convention is <m:math id="id59467" display="inline">
													<m:semantics id="id59470">
														<m:mrow id="id59471">
															<m:mi id="id59472">d</m:mi>
															<m:mo id="id59475">⟨</m:mo>
															<m:msub id="id59477">
																<m:mi id="id59478">μ</m:mi>
																<m:mi id="id59480">G</m:mi>
															</m:msub>
															<m:mo id="id59483">,</m:mo>
															<m:mi id="id59485">ξ</m:mi>
															<m:mo id="id59487">⟩</m:mo>
															<m:mo id="id59490">=</m:mo>
															<m:mi id="id59492">ω</m:mi>
															<m:mo id="id59494">(</m:mo>
															<m:mi id="id59496">V</m:mi>
															<m:mo id="id59498">(</m:mo>
															<m:mi id="id59500">ξ</m:mi>
															<m:mo id="id59503">)</m:mo>
															<m:mo id="id59505">,</m:mo>
															<m:mo id="id59507">⋅</m:mo>
															<m:mo id="id59510">)</m:mo>
															<m:mo id="id59512">,</m:mo>
															<m:mi id="id59514" mathvariant="normal">∀</m:mi>
															<m:mi id="id59518">ξ</m:mi>
															<m:mo id="id59521">∈</m:mo>
															<m:mi id="id59523" mathvariant="fraktur">g</m:mi>
														</m:mrow>
														<m:annotation-xml id="id59528" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>, where <m:math id="id59532" display="inline">
													<m:semantics id="id59536">
														<m:mrow id="id59537">
															<m:mrow id="id59538">
																<m:mi id="id59539">V</m:mi>
																<m:mo id="id59541">⁢</m:mo>
																<m:mfenced id="id59543" open="(" close=")">
																	<m:mi id="id59548">ξ</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id59551">∈</m:mo>
															<m:mrow id="id59553">
																<m:mi id="id59554" mathvariant="normal">Γ</m:mi>
																<m:mo id="id59559">⁢</m:mo>
																<m:mfenced id="id59561" open="(" close=")">
																	<m:mrow id="id59566">
																		<m:mi id="id59568">T</m:mi>
																		<m:mo id="id59570">⁢</m:mo>
																		<m:mi id="id59572">X</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id59574" encoding="MathML-Content">
															<m:apply id="id59578">
																<m:ci id="id59579">∈</m:ci>
																<m:apply id="id59581">
																	<m:times id="id59582"/>
																	<m:ci id="id59583">V</m:ci>
																	<m:ci id="id59585">ξ</m:ci>
																</m:apply>
																<m:apply id="id59588">
																	<m:times id="id59589"/>
																	<m:ci id="id59590">Γ</m:ci>
																	<m:apply id="id59592">
																		<m:times id="id59593"/>
																		<m:ci id="id59594">T</m:ci>
																		<m:ci id="id59596">X</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> denotes the vector field
generated by the infinitesimal action of <m:math id="id59600" display="inline">
													<m:semantics id="id59604">
														<m:mi id="id59605">ξ</m:mi>
														<m:annotation-xml id="id59607" encoding="MathML-Content">
															<m:ci id="id59610">ξ</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>; see the companion
paper <cite id="id59614" class="cite">[<a id="id59619" href="#bib.bib15" title="" class="ref">15</a>]</cite> for more details);</p>
										</div>

									</li>

									<li class="item nobullet" id="I1.i5">
										<span id="id59640" class="tag">
											<m:math id="id59643" display="inline">
												<m:semantics id="id59646">
													<m:mrow id="id59647">
														<m:msub id="id59648">
															<m:mi id="id59649">μ</m:mi>
															<m:mi id="id59652">T</m:mi>
														</m:msub>
														<m:mo id="id59654">:</m:mo>
														<m:mrow id="id59656">
															<m:mi id="id59657">X</m:mi>
															<m:mo id="id59659">→</m:mo>
															<m:msup id="id59661">
																<m:mi id="id59662" mathvariant="fraktur">t</m:mi>
																<m:mo id="id59667">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id59669" encoding="MathML-Content">
														<m:apply id="id59672">
															<m:ci id="id59673">:</m:ci>
															<m:apply id="id59676">
																<m:csymbol id="id59677" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id59681">μ</m:ci>
																<m:ci id="id59684">T</m:ci>
															</m:apply>
															<m:apply id="id59686">
																<m:ci id="id59687">→</m:ci>
																<m:ci id="id59689">X</m:ci>
																<m:apply id="id59691">
																	<m:csymbol id="id59692" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id59697">t</m:ci>
																	<m:times id="id59699"/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i5.p1">
											<p id="id59708" style="" class="p">is the corresponding moment map for the
restriction of the action to <m:math id="id59715" display="inline">
													<m:semantics id="id59718">
														<m:mi id="id59719">T</m:mi>
														<m:annotation-xml id="id59721" encoding="MathML-Content">
															<m:ci id="id59725">T</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> (given by composing <m:math id="id59728" display="inline">
													<m:semantics id="id59732">
														<m:msub id="id59733">
															<m:mi id="id59734">μ</m:mi>
															<m:mi id="id59736">G</m:mi>
														</m:msub>
														<m:annotation-xml id="id59738" encoding="MathML-Content">
															<m:apply id="id59742">
																<m:csymbol id="id59743" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id59747">μ</m:ci>
																<m:ci id="id59750">G</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> with the
natural projection <m:math id="id59753" display="inline">
													<m:semantics id="id59757">
														<m:mrow id="id59758">
															<m:msup id="id59759">
																<m:mi id="id59760" mathvariant="fraktur">g</m:mi>
																<m:mo id="id59764">*</m:mo>
															</m:msup>
															<m:mo id="id59766">↠</m:mo>
															<m:msup id="id59769">
																<m:mi id="id59770" mathvariant="fraktur">t</m:mi>
																<m:mo id="id59774">*</m:mo>
															</m:msup>
														</m:mrow>
														<m:annotation-xml id="id59776" encoding="MathML-Content">
															<m:apply id="id59780">
																<m:ci id="id59781">↠</m:ci>
																<m:apply id="id59783">
																	<m:csymbol id="id59784" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id59789">g</m:ci>
																	<m:times id="id59791"/>
																</m:apply>
																<m:apply id="id59792">
																	<m:csymbol id="id59793" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id59798">t</m:ci>
																	<m:times id="id59800"/>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>);</p>
										</div>

									</li>

									<li class="item nobullet" id="I1.i6">
										<span id="id59812" class="tag">
											<m:math id="id59815" display="inline">
												<m:semantics id="id59818">
													<m:mrow id="id59819">
														<m:mi id="id59820">X</m:mi>
														<m:mo id="id59823">/</m:mo>
														<m:mo id="id59825">/</m:mo>
														<m:mi id="id59827">G</m:mi>
														<m:mo id="id59829">,</m:mo>
														<m:mi id="id59831">X</m:mi>
														<m:mo id="id59833">/</m:mo>
														<m:mo id="id59835">/</m:mo>
														<m:mi id="id59838">T</m:mi>
													</m:mrow>
													<m:annotation-xml id="id59840" encoding="MathML-Content">
														<m:ci id="id59843">X</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</span>
										<div class="para" id="I1.i6.p1">
											<p id="id59853" style="" class="p">denote the symplectic quotients
												<m:math id="id59859" display="inline">
													<m:semantics id="id59863">
														<m:mrow id="id59864">
															<m:mrow id="id59865">
																<m:msubsup id="id59866">
																	<m:mi id="id59867">μ</m:mi>
																	<m:mi id="id59869">G</m:mi>
																	<m:mrow id="id59872">
																		<m:mo id="id59873">-</m:mo>
																		<m:mn id="id59875">1</m:mn>
																	</m:mrow>
																</m:msubsup>
																<m:mo id="id59877">⁢</m:mo>
																<m:mfenced id="id59879" open="(" close=")">
																	<m:mn id="id59884">0</m:mn>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id59886">/</m:mo>
															<m:mi id="id59889">G</m:mi>
														</m:mrow>
														<m:annotation-xml id="id59891" encoding="MathML-Content">
															<m:apply id="id59894">
																<m:divide id="id59895"/>
																<m:apply id="id59896">
																	<m:times id="id59897"/>
																	<m:apply id="id59898">
																		<m:csymbol id="id59899" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id59904">
																			<m:csymbol id="id59905" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id59910">μ</m:ci>
																			<m:ci id="id59912">G</m:ci>
																		</m:apply>
																		<m:apply id="id59914">
																			<m:minus id="id59915"/>
																			<m:cn id="id59916">1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn id="id59919">0</m:cn>
																</m:apply>
																<m:ci id="id59921">G</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> and <m:math id="id59924" display="inline">
													<m:semantics id="id59928">
														<m:mrow id="id59929">
															<m:mrow id="id59930">
																<m:msubsup id="id59931">
																	<m:mi id="id59932">μ</m:mi>
																	<m:mi id="id59934">T</m:mi>
																	<m:mrow id="id59936">
																		<m:mo id="id59937">-</m:mo>
																		<m:mn id="id59940">1</m:mn>
																	</m:mrow>
																</m:msubsup>
																<m:mo id="id59942">⁢</m:mo>
																<m:mfenced id="id59944" open="(" close=")">
																	<m:mn id="id59949">0</m:mn>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id59951">/</m:mo>
															<m:mi id="id59953">T</m:mi>
														</m:mrow>
														<m:annotation-xml id="id59956" encoding="MathML-Content">
															<m:apply id="id59959">
																<m:divide id="id59960"/>
																<m:apply id="id59961">
																	<m:times id="id59962"/>
																	<m:apply id="id59963">
																		<m:csymbol id="id59964" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id59969">
																			<m:csymbol id="id59970" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id59975">μ</m:ci>
																			<m:ci id="id59977">T</m:ci>
																		</m:apply>
																		<m:apply id="id59979">
																			<m:minus id="id59980"/>
																			<m:cn id="id59981">1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn id="id59983">0</m:cn>
																</m:apply>
																<m:ci id="id59986">T</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> respectively.</p>
										</div>

									</li>

								</ul>
							</div>

						</div>

					</div>

					<div class="section" id="S1">
						<h2 id="id60003" class="title section-title">
							<span id="id60006" class="tag">1 </span>The main topological result</h2>
						<div class="para" id="S1.p1">
							<p id="id60020" style="" class="p">The results of this paper all follow from one topological result,
which we prove in this section. This result is stated in terms of
certain complex line bundles on <m:math id="id60027" display="inline">
									<m:semantics id="id60030">
										<m:mrow id="id60032">
											<m:mi id="id60033">X</m:mi>
											<m:mo id="id60035">/</m:mo>
											<m:mo id="id60037">/</m:mo>
											<m:mi id="id60039">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60041" encoding="MathML-Content">
											<m:ci id="id60044">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>:</p>
						</div>

						<div class="theorem definition" id="S1.E1">
							<h6 id="id60056" class="title theorem-title bold">
								<span id="id60060" class="tag">Definition 1.1</span>.</h6>
							<div class="para" id="S1.E1.p1">
								<p id="id60073" style="" class="p">Let <m:math id="id60080" display="inline">
										<m:semantics id="id60083">
											<m:mi id="id60084">α</m:mi>
											<m:annotation-xml id="id60086" encoding="MathML-Content">
												<m:ci id="id60090">α</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> be a weight of <m:math id="id60093" display="inline">
										<m:semantics id="id60097">
											<m:mi id="id60098">T</m:mi>
											<m:annotation-xml id="id60100" encoding="MathML-Content">
												<m:ci id="id60103">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, that is, a one-dimensional
representation, and let <m:math id="id60107" display="inline">
										<m:semantics id="id60110">
											<m:msub id="id60111">
												<m:mi id="id60112" mathvariant="double-struck">C</m:mi>
												<m:mfenced id="id60117" open="(" close=")">
													<m:mi id="id60122">α</m:mi>
												</m:mfenced>
											</m:msub>
											<m:annotation-xml id="id60124" encoding="MathML-Content">
												<m:apply id="id60128">
													<m:csymbol id="id60129" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id60133">C</m:ci>
													<m:ci id="id60136">α</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> denote the corresponding
representation space. Then we define the line bundle <m:math id="id60140" display="inline">
										<m:semantics id="id60143">
											<m:mrow id="id60144">
												<m:msub id="id60145">
													<m:mi id="id60146">L</m:mi>
													<m:mi id="id60148">α</m:mi>
												</m:msub>
												<m:mo id="id60151">→</m:mo>
												<m:mi id="id60153">X</m:mi>
												<m:mo id="id60155">/</m:mo>
												<m:mo id="id60158">/</m:mo>
												<m:mi id="id60160">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id60162" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>
to be the associated bundle</p>
								<table class="equation" id="S1.Ex1">

									<tr valign="baseline" class="equation" id="S1.Ex1">
										<td id="id60183" class="eqpad"/>
										<td id="id60188" nowrap="yes" align="center" colspan="1">
											<m:math id="id60195" display="block">
												<m:semantics id="id60198">
													<m:mrow id="id60199">
														<m:msub id="id60200">
															<m:mi id="id60202">L</m:mi>
															<m:mi id="id60204">α</m:mi>
														</m:msub>
														<m:mo id="id60206">:=</m:mo>
														<m:msubsup id="id60208">
															<m:mi id="id60209">μ</m:mi>
															<m:mi id="id60212">T</m:mi>
															<m:mrow id="id60214">
																<m:mo id="id60215">-</m:mo>
																<m:mn id="id60217">1</m:mn>
															</m:mrow>
														</m:msubsup>
														<m:mo id="id60219">(</m:mo>
														<m:mn id="id60221">0</m:mn>
														<m:mo id="id60223">)</m:mo>
														<m:msub id="id60226">
															<m:mo id="id60227">×</m:mo>
															<m:mi id="id60229">T</m:mi>
														</m:msub>
														<m:msub id="id60231">
															<m:mi id="id60232" mathvariant="double-struck">C</m:mi>
															<m:mfenced id="id60237" open="(" close=")">
																<m:mi id="id60242">α</m:mi>
															</m:mfenced>
														</m:msub>
														<m:mo id="id60244">[</m:mo>
														<m:mi id="id60246">d</m:mi>
														<m:mo id="id60248">]</m:mo>
														<m:mi id="id60250">X</m:mi>
														<m:mo id="id60253">/</m:mo>
														<m:mo id="id60255">/</m:mo>
														<m:mi id="id60257">T</m:mi>
													</m:mrow>
													<m:annotation-xml id="id60259" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id60263" class="eqpad"/>
									</tr>

								</table>
							</div>

						</div>

						<div class="para" id="S1.p2">
							<p id="id60278" style="" class="p">We denote by <m:math id="id60284" display="inline">
									<m:semantics id="id60287">
										<m:mi id="id60288" mathvariant="normal">Δ</m:mi>
										<m:annotation-xml id="id60293" encoding="MathML-Content">
											<m:ci id="id60296">Δ</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> the set of roots of <m:math id="id60300" display="inline">
									<m:semantics id="id60304">
										<m:mi id="id60305">G</m:mi>
										<m:annotation-xml id="id60307" encoding="MathML-Content">
											<m:ci id="id60310">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, that is, <m:math id="id60314" display="inline">
									<m:semantics id="id60317">
										<m:mi id="id60318" mathvariant="normal">Δ</m:mi>
										<m:annotation-xml id="id60323" encoding="MathML-Content">
											<m:ci id="id60326">Δ</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is
the set of nonzero weights which occur in the action of <m:math id="id60330" display="inline">
									<m:semantics id="id60333">
										<m:mi id="id60334">T</m:mi>
										<m:annotation-xml id="id60337" encoding="MathML-Content">
											<m:ci id="id60340">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> on
								<m:math id="id60343" display="inline">
									<m:semantics id="id60347">
										<m:mrow id="id60348">
											<m:mi id="id60349" mathvariant="fraktur">g</m:mi>
											<m:mo id="id60353">⊗</m:mo>
											<m:mi id="id60356" mathvariant="double-struck">C</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60360" encoding="MathML-Content">
											<m:apply id="id60363">
												<m:ci id="id60364">⊗</m:ci>
												<m:ci id="id60367">g</m:ci>
												<m:ci id="id60369">C</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>; we fix a choice <m:math id="id60372" display="inline">
									<m:semantics id="id60376">
										<m:mrow id="id60377">
											<m:msup id="id60378">
												<m:mi id="id60379" mathvariant="normal">Δ</m:mi>
												<m:mo id="id60384">+</m:mo>
											</m:msup>
											<m:mo id="id60386">⊂</m:mo>
											<m:mi id="id60388" mathvariant="normal">Δ</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60393" encoding="MathML-Content">
											<m:apply id="id60396">
												<m:ci id="id60397">⊂</m:ci>
												<m:apply id="id60400">
													<m:csymbol id="id60401" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id60405">Δ</m:ci>
													<m:plus id="id60408"/>
												</m:apply>
												<m:ci id="id60409">Δ</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of positive
roots, and denote by <m:math id="id60413" display="inline">
									<m:semantics id="id60416">
										<m:msup id="id60417">
											<m:mi id="id60418" mathvariant="normal">Δ</m:mi>
											<m:mo id="id60423">-</m:mo>
										</m:msup>
										<m:annotation-xml id="id60425" encoding="MathML-Content">
											<m:apply id="id60428">
												<m:csymbol id="id60429" cd="ambiguous">superscript</m:csymbol>
												<m:ci id="id60434">Δ</m:ci>
												<m:minus id="id60436"/>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> the corresponding negative roots.</p>
						</div>

						<div class="para" id="S1.p3">
							<p id="id60448" style="" class="p">
								<m:math id="id60453" display="inline">
									<m:semantics id="id60456">
										<m:mrow id="id60457">
											<m:mi id="id60458">X</m:mi>
											<m:mo id="id60460">/</m:mo>
											<m:mo id="id60463">/</m:mo>
											<m:mi id="id60465">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60467" encoding="MathML-Content">
											<m:ci id="id60470">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id60474" display="inline">
									<m:semantics id="id60477">
										<m:mrow id="id60478">
											<m:mi id="id60479">X</m:mi>
											<m:mo id="id60481">/</m:mo>
											<m:mo id="id60483">/</m:mo>
											<m:mi id="id60486">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60488" encoding="MathML-Content">
											<m:ci id="id60491">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are related by
a fibering and an inclusion:</p>
							<table class="equation" id="S1.Ex2">

								<tr valign="baseline" class="equation" id="S1.Ex2">
									<td id="id60511" class="eqpad"/>
									<td id="id60516" nowrap="yes" align="center" colspan="1">
										<m:math id="id60523" display="block">
											<m:semantics id="id60526">
												<m:mrow id="id60528">
													<m:msubsup id="id60529">
														<m:mi id="id60530">μ</m:mi>
														<m:mi id="id60532">G</m:mi>
														<m:mrow id="id60534">
															<m:mo id="id60535">-</m:mo>
															<m:mn id="id60537">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id60540">(</m:mo>
													<m:mn id="id60542">0</m:mn>
													<m:mo id="id60544">)</m:mo>
													<m:mo id="id60546">/</m:mo>
													<m:mi id="id60548">T</m:mi>
													<m:mi id="id60550" mathvariant="normal">@</m:mi>
													<m:mmultiscripts id="id60555">
														<m:mo id="id60556">-</m:mo>
														<m:mprescripts id="id60559"/>
														<m:none id="id60560"/>
														<m:mo id="id60561">(</m:mo>
													</m:mmultiscripts>
													<m:mo id="id60563">&gt;</m:mo>
													<m:mo id="id60565">[</m:mo>
													<m:mi id="id60567">r</m:mi>
													<m:msup id="id60570">
														<m:mo id="id60571">]</m:mo>
														<m:mi id="id60573">i</m:mi>
													</m:msup>
													<m:mo id="id60575">[</m:mo>
													<m:mi id="id60577">d</m:mi>
													<m:msup id="id60579">
														<m:mo id="id60580">]</m:mo>
														<m:mi id="id60582">π</m:mi>
													</m:msup>
													<m:msubsup id="id60585">
														<m:mi id="id60586">μ</m:mi>
														<m:mi id="id60588">T</m:mi>
														<m:mrow id="id60590">
															<m:mo id="id60591">-</m:mo>
															<m:mn id="id60594">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id60596">(</m:mo>
													<m:mn id="id60598">0</m:mn>
													<m:mo id="id60600">)</m:mo>
													<m:mo id="id60602">/</m:mo>
													<m:mi id="id60604">T</m:mi>
													<m:mo id="id60606">=</m:mo>
													<m:mi id="id60608">X</m:mi>
													<m:mo id="id60611">/</m:mo>
													<m:mo id="id60613">/</m:mo>
													<m:mi id="id60615">T</m:mi>
													<m:mo id="id60617">*</m:mo>
													<m:mo id="id60619">*</m:mo>
													<m:mo id="id60621">[</m:mo>
													<m:mi id="id60623">l</m:mi>
													<m:mo id="id60626">]</m:mo>
													<m:mi id="id60628">X</m:mi>
													<m:mo id="id60630">/</m:mo>
													<m:mo id="id60632">/</m:mo>
													<m:mi id="id60634">G</m:mi>
													<m:mo id="id60636">=</m:mo>
													<m:msubsup id="id60638">
														<m:mi id="id60639">μ</m:mi>
														<m:mi id="id60642">G</m:mi>
														<m:mrow id="id60644">
															<m:mo id="id60645">-</m:mo>
															<m:mn id="id60647">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id60649">(</m:mo>
													<m:mn id="id60651">0</m:mn>
													<m:mo id="id60654">)</m:mo>
													<m:mo id="id60656">/</m:mo>
													<m:mi id="id60658">G</m:mi>
													<m:mo id="id60660">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id60662" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</td>
									<td id="id60666" class="eqpad"/>
								</tr>

							</table>
							<p id="id60673" style="" class="p">Note that <m:math id="id60679" display="inline">
									<m:semantics id="id60683">
										<m:mrow id="id60684">
											<m:mi id="id60685">X</m:mi>
											<m:mo id="id60687">/</m:mo>
											<m:mo id="id60689">/</m:mo>
											<m:mi id="id60691">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60693" encoding="MathML-Content">
											<m:ci id="id60697">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id60700" display="inline">
									<m:semantics id="id60704">
										<m:mrow id="id60705">
											<m:mi id="id60706">X</m:mi>
											<m:mo id="id60708">/</m:mo>
											<m:mo id="id60710">/</m:mo>
											<m:mi id="id60712">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id60714" encoding="MathML-Content">
											<m:ci id="id60718">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are symplectic manifolds, and hence
possess compactible almost complex structures, unique up to
homotopy <cite id="id60723" class="cite">[<a id="id60727" href="#bib.bib17" title="" class="ref">17</a>, proposition 4.1]</cite>.</p>
						</div>

						<div class="theorem proposition" id="S1.E2">
							<h6 id="id60747" class="title theorem-title bold">
								<span id="id60750" class="tag">Proposition 1.2</span>.</h6>
							<div class="para" id="S1.E2.p1">
								<ol class="enumerate" id="I2">

									<li class="item" id="I2.i1">

										<div class="para" id="I2.i1.p1">
											<p id="id60784" style="" class="p">The vector bundle <m:math id="id60790" display="inline">
													<m:semantics id="id60794">
														<m:mrow id="id60795">
															<m:msub id="id60796">
																<m:mo id="id60797">⊕</m:mo>
																<m:mrow id="id60799">
																	<m:mi id="id60800">α</m:mi>
																	<m:mo id="id60803">∈</m:mo>
																	<m:msup id="id60805">
																		<m:mi id="id60806" mathvariant="normal">Δ</m:mi>
																		<m:mo id="id60811">-</m:mo>
																	</m:msup>
																</m:mrow>
															</m:msub>
															<m:msub id="id60813">
																<m:mi id="id60814">L</m:mi>
																<m:mi id="id60816">α</m:mi>
															</m:msub>
															<m:mo id="id60818">→</m:mo>
															<m:mi id="id60821">X</m:mi>
															<m:mo id="id60823">/</m:mo>
															<m:mo id="id60825">/</m:mo>
															<m:mi id="id60827">T</m:mi>
														</m:mrow>
														<m:annotation-xml id="id60829" encoding="MathML-Content"/>
													</m:semantics>
												</m:math> has a section <m:math id="id60834" display="inline">
													<m:semantics id="id60837">
														<m:mi id="id60838">s</m:mi>
														<m:annotation-xml id="id60841" encoding="MathML-Content">
															<m:ci id="id60844">s</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, which is
transverse to the zero section, and such that the zeroset of <m:math id="id60848" display="inline">
													<m:semantics id="id60851">
														<m:mi id="id60852">s</m:mi>
														<m:annotation-xml id="id60854" encoding="MathML-Content">
															<m:ci id="id60858">s</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> is
the submanifold <m:math id="id60861" display="inline">
													<m:semantics id="id60865">
														<m:mrow id="id60866">
															<m:msubsup id="id60867">
																<m:mi id="id60868">μ</m:mi>
																<m:mi id="id60870">G</m:mi>
																<m:mrow id="id60872">
																	<m:mo id="id60873">-</m:mo>
																	<m:mn id="id60876">1</m:mn>
																</m:mrow>
															</m:msubsup>
															<m:mo id="id60878">(</m:mo>
															<m:mn id="id60880">0</m:mn>
															<m:mo id="id60882">)</m:mo>
															<m:mo id="id60884">/</m:mo>
															<m:mi id="id60886">T</m:mi>
															<m:mo id="id60888">⊂</m:mo>
															<m:mi id="id60891">X</m:mi>
															<m:mo id="id60893">/</m:mo>
															<m:mo id="id60895">/</m:mo>
															<m:mi id="id60897">T</m:mi>
														</m:mrow>
														<m:annotation-xml id="id60899" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>. It follows that the derivative
of <m:math id="id60904" display="inline">
													<m:semantics id="id60907">
														<m:mi id="id60908">s</m:mi>
														<m:annotation-xml id="id60911" encoding="MathML-Content">
															<m:ci id="id60914">s</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> identifies the normal bundle</p>
											<table class="equation" id="S1.Ex3">

												<tr valign="baseline" class="equation" id="S1.Ex3">
													<td id="id60934" class="eqpad"/>
													<td id="id60939" nowrap="yes" align="center" colspan="1">
														<m:math id="id60946" display="block">
															<m:semantics id="id60949">
																<m:mrow id="id60950">
																	<m:mrow id="id60951">
																		<m:mrow id="id60952">
																			<m:mi id="id60954">ν</m:mi>
																			<m:mo id="id60956">⁢</m:mo>
																			<m:mfenced id="id60958" open="(" close=")">
																				<m:mrow id="id60963">
																					<m:mrow id="id60964">
																						<m:msubsup id="id60966">
																							<m:mi id="id60967">μ</m:mi>
																							<m:mi id="id60969">G</m:mi>
																							<m:mrow id="id60971">
																								<m:mo id="id60972">-</m:mo>
																								<m:mn id="id60974">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id60976">⁢</m:mo>
																						<m:mfenced id="id60979" open="(" close=")">
																							<m:mn id="id60984">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																					<m:mo id="id60986">/</m:mo>
																					<m:mi id="id60988">T</m:mi>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																		<m:mo id="id60990">≅</m:mo>
																		<m:mrow id="id60993">
																			<m:msub id="id60994">
																				<m:mo id="id60995">⊕</m:mo>
																				<m:mrow id="id60997">
																					<m:mi id="id60998">α</m:mi>
																					<m:mo id="id61001">∈</m:mo>
																					<m:msup id="id61003">
																						<m:mi id="id61004" mathvariant="normal">Δ</m:mi>
																						<m:mo id="id61009">-</m:mo>
																					</m:msup>
																				</m:mrow>
																			</m:msub>
																			<m:mmultiscripts id="id61011">
																				<m:mi id="id61012">L</m:mi>
																				<m:mi id="id61014">α</m:mi>
																				<m:none id="id61017"/>
																				<m:mrow id="id61018">
																					<m:mrow id="id61019">
																						<m:msubsup id="id61020">
																							<m:mi id="id61021">μ</m:mi>
																							<m:mi id="id61023">G</m:mi>
																							<m:mrow id="id61025">
																								<m:mo id="id61026">-</m:mo>
																								<m:mn id="id61029">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id61031">⁢</m:mo>
																						<m:mfenced id="id61033" open="(" close=")">
																							<m:mn id="id61038">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																					<m:mo id="id61040">/</m:mo>
																					<m:mi id="id61042">T</m:mi>
																				</m:mrow>
																				<m:none id="id61045"/>
																			</m:mmultiscripts>
																		</m:mrow>
																	</m:mrow>
																	<m:mo id="id61046">.</m:mo>
																</m:mrow>
																<m:annotation-xml id="id61048" encoding="MathML-Content">
																	<m:apply id="id61051">
																		<m:approx id="id61052"/>
																		<m:apply id="id61053">
																			<m:times id="id61054"/>
																			<m:ci id="id61055">ν</m:ci>
																			<m:apply id="id61058">
																				<m:divide id="id61059"/>
																				<m:apply id="id61060">
																					<m:times id="id61061"/>
																					<m:apply id="id61062">
																						<m:csymbol id="id61063" cd="ambiguous">superscript</m:csymbol>
																						<m:apply id="id61068">
																							<m:csymbol id="id61069" cd="ambiguous">subscript</m:csymbol>
																							<m:ci id="id61074">μ</m:ci>
																							<m:ci id="id61076">G</m:ci>
																						</m:apply>
																						<m:apply id="id61078">
																							<m:minus id="id61079"/>
																							<m:cn id="id61080">1</m:cn>
																						</m:apply>
																					</m:apply>
																					<m:cn id="id61082">0</m:cn>
																				</m:apply>
																				<m:ci id="id61084">T</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id61087">
																			<m:apply id="id61088">
																				<m:csymbol id="id61089" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id61093">⊕</m:ci>
																				<m:apply id="id61096">
																					<m:ci id="id61097">∈</m:ci>
																					<m:ci id="id61099">α</m:ci>
																					<m:apply id="id61102">
																						<m:csymbol id="id61103" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id61107">Δ</m:ci>
																						<m:minus id="id61110"/>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id61111">
																				<m:csymbol id="id61112" cd="ambiguous">subscript</m:csymbol>
																				<m:apply id="id61117">
																					<m:csymbol id="id61118" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id61122">L</m:ci>
																					<m:ci id="id61124">α</m:ci>
																				</m:apply>
																				<m:apply id="id61127">
																					<m:divide id="id61128"/>
																					<m:apply id="id61129">
																						<m:times id="id61130"/>
																						<m:apply id="id61131">
																							<m:csymbol id="id61132" cd="ambiguous">superscript</m:csymbol>
																							<m:apply id="id61137">
																								<m:csymbol id="id61138" cd="ambiguous">subscript</m:csymbol>
																								<m:ci id="id61143">μ</m:ci>
																								<m:ci id="id61145">G</m:ci>
																							</m:apply>
																							<m:apply id="id61147">
																								<m:minus id="id61148"/>
																								<m:cn id="id61149">1</m:cn>
																							</m:apply>
																						</m:apply>
																						<m:cn id="id61151">0</m:cn>
																					</m:apply>
																					<m:ci id="id61154">T</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:annotation-xml>
															</m:semantics>
														</m:math>
													</td>
													<td id="id61157" class="eqpad"/>
												</tr>

											</table>
										</div>

									</li>

									<li class="item" id="I2.i2">

										<div class="para" id="I2.i2.p1">
											<p id="id61178" style="" class="p">Letting <m:math id="id61184" display="inline">
													<m:semantics id="id61187">
														<m:mrow id="id61188">
															<m:mrow id="id61190">
																<m:mrow id="id61191">
																	<m:mi id="id61192">v</m:mi>
																	<m:mo id="id61194">⁢</m:mo>
																	<m:mi id="id61196">e</m:mi>
																	<m:mo id="id61198">⁢</m:mo>
																	<m:mi id="id61201">r</m:mi>
																	<m:mo id="id61203">⁢</m:mo>
																	<m:mi id="id61205">t</m:mi>
																</m:mrow>
																<m:mo id="id61207">⁢</m:mo>
																<m:mfenced id="id61210" open="(" close=")">
																	<m:mi id="id61215">π</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id61217">→</m:mo>
															<m:mrow id="id61220">
																<m:mrow id="id61221">
																	<m:msubsup id="id61222">
																		<m:mi id="id61223">μ</m:mi>
																		<m:mi id="id61225">G</m:mi>
																		<m:mrow id="id61227">
																			<m:mo id="id61228">-</m:mo>
																			<m:mn id="id61231">1</m:mn>
																		</m:mrow>
																	</m:msubsup>
																	<m:mo id="id61233">⁢</m:mo>
																	<m:mfenced id="id61235" open="(" close=")">
																		<m:mn id="id61240">0</m:mn>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id61242">/</m:mo>
																<m:mi id="id61244">T</m:mi>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id61247" encoding="MathML-Content">
															<m:apply id="id61250">
																<m:ci id="id61251">→</m:ci>
																<m:apply id="id61253">
																	<m:times id="id61254"/>
																	<m:ci id="id61256"/>
																	<m:ci id="id61257">π</m:ci>
																</m:apply>
																<m:apply id="id61259">
																	<m:divide id="id61260"/>
																	<m:apply id="id61261">
																		<m:times id="id61262"/>
																		<m:apply id="id61263">
																			<m:csymbol id="id61264" cd="ambiguous">superscript</m:csymbol>
																			<m:apply id="id61269">
																				<m:csymbol id="id61270" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id61275">μ</m:ci>
																				<m:ci id="id61277">G</m:ci>
																			</m:apply>
																			<m:apply id="id61279">
																				<m:minus id="id61280"/>
																				<m:cn id="id61281">1</m:cn>
																			</m:apply>
																		</m:apply>
																		<m:cn id="id61284">0</m:cn>
																	</m:apply>
																	<m:ci id="id61286">T</m:ci>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> denote the vector bundle of tangents to
the fibres of <m:math id="id61289" display="inline">
													<m:semantics id="id61293">
														<m:mi id="id61294">π</m:mi>
														<m:annotation-xml id="id61296" encoding="MathML-Content">
															<m:ci id="id61300">π</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, we have</p>
											<table class="equation" id="S1.Ex4">

												<tr valign="baseline" class="equation" id="S1.Ex4">
													<td id="id61320" class="eqpad"/>
													<td id="id61324" nowrap="yes" align="center" colspan="1">
														<m:math id="id61332" display="block">
															<m:semantics id="id61335">
																<m:mrow id="id61336">
																	<m:mrow id="id61337">
																		<m:mrow id="id61338">
																			<m:mrow id="id61339">
																				<m:mi id="id61340">v</m:mi>
																				<m:mo id="id61342">⁢</m:mo>
																				<m:mi id="id61345">e</m:mi>
																				<m:mo id="id61347">⁢</m:mo>
																				<m:mi id="id61349">r</m:mi>
																				<m:mo id="id61352">⁢</m:mo>
																				<m:mi id="id61354">t</m:mi>
																			</m:mrow>
																			<m:mo id="id61356">⁢</m:mo>
																			<m:mfenced id="id61358" open="(" close=")">
																				<m:mi id="id61364">π</m:mi>
																			</m:mfenced>
																		</m:mrow>
																		<m:mo id="id61366">≅</m:mo>
																		<m:mrow id="id61368">
																			<m:msub id="id61369">
																				<m:mo id="id61370">⊕</m:mo>
																				<m:mrow id="id61373">
																					<m:mi id="id61374">α</m:mi>
																					<m:mo id="id61376">∈</m:mo>
																					<m:msup id="id61379">
																						<m:mi id="id61380" mathvariant="normal">Δ</m:mi>
																						<m:mo id="id61384">+</m:mo>
																					</m:msup>
																				</m:mrow>
																			</m:msub>
																			<m:mmultiscripts id="id61387">
																				<m:mi id="id61388">L</m:mi>
																				<m:mi id="id61390">α</m:mi>
																				<m:none id="id61392"/>
																				<m:mrow id="id61393">
																					<m:mrow id="id61394">
																						<m:msubsup id="id61395">
																							<m:mi id="id61396">μ</m:mi>
																							<m:mi id="id61399">G</m:mi>
																							<m:mrow id="id61401">
																								<m:mo id="id61402">-</m:mo>
																								<m:mn id="id61404">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id61406">⁢</m:mo>
																						<m:mfenced id="id61409" open="(" close=")">
																							<m:mn id="id61414">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																					<m:mo id="id61416">/</m:mo>
																					<m:mi id="id61418">T</m:mi>
																				</m:mrow>
																				<m:none id="id61420"/>
																			</m:mmultiscripts>
																		</m:mrow>
																	</m:mrow>
																	<m:mo id="id61421">.</m:mo>
																</m:mrow>
																<m:annotation-xml id="id61423" encoding="MathML-Content">
																	<m:apply id="id61427">
																		<m:approx id="id61428"/>
																		<m:apply id="id61429">
																			<m:times id="id61430"/>
																			<m:ci id="id61431"/>
																			<m:ci id="id61432">π</m:ci>
																		</m:apply>
																		<m:apply id="id61434">
																			<m:apply id="id61436">
																				<m:csymbol id="id61437" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id61441">⊕</m:ci>
																				<m:apply id="id61444">
																					<m:ci id="id61445">∈</m:ci>
																					<m:ci id="id61447">α</m:ci>
																					<m:apply id="id61450">
																						<m:csymbol id="id61451" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id61455">Δ</m:ci>
																						<m:plus id="id61458"/>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id61459">
																				<m:csymbol id="id61460" cd="ambiguous">subscript</m:csymbol>
																				<m:apply id="id61464">
																					<m:csymbol id="id61466" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id61470">L</m:ci>
																					<m:ci id="id61472">α</m:ci>
																				</m:apply>
																				<m:apply id="id61475">
																					<m:divide id="id61476"/>
																					<m:apply id="id61477">
																						<m:times id="id61478"/>
																						<m:apply id="id61479">
																							<m:csymbol id="id61480" cd="ambiguous">superscript</m:csymbol>
																							<m:apply id="id61485">
																								<m:csymbol id="id61486" cd="ambiguous">subscript</m:csymbol>
																								<m:ci id="id61490">μ</m:ci>
																								<m:ci id="id61493">G</m:ci>
																							</m:apply>
																							<m:apply id="id61495">
																								<m:minus id="id61496"/>
																								<m:cn id="id61497">1</m:cn>
																							</m:apply>
																						</m:apply>
																						<m:cn id="id61499">0</m:cn>
																					</m:apply>
																					<m:ci id="id61501">T</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:annotation-xml>
															</m:semantics>
														</m:math>
													</td>
													<td id="id61505" class="eqpad"/>
												</tr>

											</table>
										</div>

									</li>

									<li class="item" id="I2.i3">

										<div class="para" id="I2.i3.p1">
											<p id="id61526" style="" class="p">There is a complex orientation<span id="id61532" class="note footnote">
													<sup id="id61535" class="mark">5</sup>
													<div id="id61540" class="note_content">
														<sup id="id61543" class="mark">5</sup>an almost complex structure on
a manifold or a vector bundle defines a complex orientation, and two
almost complex structures which are homotopic (through almost complex
structures) define the same complex orientation. For the definition of
complex orientation (which also involves stabilization)
see <cite id="id61554" class="cite">[<a id="id61558" href="#bib.bib20" title="" class="ref">20</a>]</cite>.
													</div>
												</span>of <m:math id="id61570" display="inline">
													<m:semantics id="id61574">
														<m:mrow id="id61575">
															<m:mrow id="id61576">
																<m:msubsup id="id61577">
																	<m:mi id="id61578">μ</m:mi>
																	<m:mi id="id61580">G</m:mi>
																	<m:mrow id="id61582">
																		<m:mo id="id61583">-</m:mo>
																		<m:mn id="id61586">1</m:mn>
																	</m:mrow>
																</m:msubsup>
																<m:mo id="id61588">⁢</m:mo>
																<m:mfenced id="id61590" open="(" close=")">
																	<m:mn id="id61595">0</m:mn>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id61597">/</m:mo>
															<m:mi id="id61599">T</m:mi>
														</m:mrow>
														<m:annotation-xml id="id61602" encoding="MathML-Content">
															<m:apply id="id61605">
																<m:divide id="id61606"/>
																<m:apply id="id61607">
																	<m:times id="id61608"/>
																	<m:apply id="id61609">
																		<m:csymbol id="id61610" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id61615">
																			<m:csymbol id="id61616" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id61621">μ</m:ci>
																			<m:ci id="id61623">G</m:ci>
																		</m:apply>
																		<m:apply id="id61625">
																			<m:minus id="id61626"/>
																			<m:cn id="id61627">1</m:cn>
																		</m:apply>
																	</m:apply>
																	<m:cn id="id61629">0</m:cn>
																</m:apply>
																<m:ci id="id61632">T</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> such that the above
isomorphisms are isomorphisms of complex-oriented spaces and
vector bundles, with respect to the complex orientations of <m:math id="id61636" display="inline">
													<m:semantics id="id61639">
														<m:mrow id="id61640">
															<m:mi id="id61642">X</m:mi>
															<m:mo id="id61644">/</m:mo>
															<m:mo id="id61646">/</m:mo>
															<m:mi id="id61648">G</m:mi>
														</m:mrow>
														<m:annotation-xml id="id61650" encoding="MathML-Content">
															<m:ci id="id61653">X</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
and <m:math id="id61657" display="inline">
													<m:semantics id="id61660">
														<m:mrow id="id61661">
															<m:mi id="id61662">X</m:mi>
															<m:mo id="id61664">/</m:mo>
															<m:mo id="id61667">/</m:mo>
															<m:mi id="id61669">T</m:mi>
														</m:mrow>
														<m:annotation-xml id="id61671" encoding="MathML-Content">
															<m:ci id="id61674">X</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> induced by their symplectic forms.</p>
										</div>

									</li>

								</ol>
							</div>

						</div>

						<div class="para" id="S1.p4">
							<p id="id61691" style="" class="p">A complex orientation induces a real orientation in a standard manner,
and for most of this paper we will only need that the above
isomorphisms are isomorphisms of real-oriented spaces and vector
bundles. We will need complex orientations for the results on
indices of elliptic operators and characteristic numbers, in
sections <a id="id61704" href="#S4" title="4 The index formula ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id61712" style="" class="text tag">4</span>
								</a> and <a id="id61720" href="#S5" title="5 Characteristic numbers ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id61718" style="" class="text tag">5</span>
								</a>.</p>
						</div>

						<div id="id61736" class="proof">
							<h6 id="id61741" class="title proof-title italic">Proof.</h6>
							<div class="para" id="S1.p5">
								<p id="id61753" style="" class="p">We prove the three statements in the proposition in
order.</p>
							</div>

							<div class="para" id="S1.p6">
								<p id="id61768" style="" class="p">
									<span id="id61774" style="" class="text italic">1. The inclusion.</span>
We have the commuting triangle of maps
								</p>
								<table class="equation" id="S1.Ex5">

									<tr valign="baseline" class="equation" id="S1.Ex5">
										<td id="id61798" class="eqpad"/>
										<td id="id61803" nowrap="yes" align="center" colspan="1">
											<m:math id="id61810" display="block">
												<m:semantics id="id61814">
													<m:mrow id="id61815">
														<m:mrow id="id61816">
															<m:mi id="id61817">X</m:mi>
															<m:mo id="id61819">⁢</m:mo>
															<m:msup id="id61821">
																<m:mfenced id="id61822" open="[" close="]">
																	<m:mi id="id61827">r</m:mi>
																</m:mfenced>
																<m:msub id="id61830">
																	<m:mi id="id61831">μ</m:mi>
																	<m:mi id="id61833">G</m:mi>
																</m:msub>
															</m:msup>
															<m:mo id="id61835">⁢</m:mo>
															<m:msub id="id61838">
																<m:mfenced id="id61839" open="[" close="]">
																	<m:mrow id="id61844">
																		<m:mi id="id61845">r</m:mi>
																		<m:mo id="id61847">⁢</m:mo>
																		<m:mi id="id61849">d</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:msub id="id61851">
																	<m:mi id="id61852">μ</m:mi>
																	<m:mi id="id61855">T</m:mi>
																</m:msub>
															</m:msub>
															<m:mo id="id61857">⁢</m:mo>
															<m:msup id="id61859">
																<m:mi id="id61860" mathvariant="fraktur">g</m:mi>
																<m:mo id="id61865">*</m:mo>
															</m:msup>
															<m:mo id="id61867">⁢</m:mo>
															<m:mi id="id61869" mathvariant="normal">@</m:mi>
														</m:mrow>
														<m:mo id="id61874">≫</m:mo>
														<m:mrow id="id61876">
															<m:msup id="id61877">
																<m:mfenced id="id61878" open="[" close="]">
																	<m:mi id="id61883">d</m:mi>
																</m:mfenced>
																<m:mi id="id61886">p</m:mi>
															</m:msup>
															<m:mo id="id61888">⁢</m:mo>
															<m:msup id="id61890">
																<m:mi id="id61891" mathvariant="fraktur">t</m:mi>
																<m:mo id="id61896">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id61898" encoding="MathML-Content">
														<m:apply id="id61901">
															<m:ci id="id61902">≫</m:ci>
															<m:apply id="id61904">
																<m:times id="id61906"/>
																<m:ci id="id61907">X</m:ci>
																<m:apply id="id61909">
																	<m:csymbol id="id61910" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id61914">r</m:ci>
																	<m:apply id="id61917">
																		<m:csymbol id="id61918" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id61922">μ</m:ci>
																		<m:ci id="id61925">G</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id61927">
																	<m:csymbol id="id61928" cd="ambiguous">subscript</m:csymbol>
																	<m:apply id="id61933">
																		<m:times id="id61934"/>
																		<m:ci id="id61935">r</m:ci>
																		<m:ci id="id61937">d</m:ci>
																	</m:apply>
																	<m:apply id="id61939">
																		<m:csymbol id="id61940" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id61945">μ</m:ci>
																		<m:ci id="id61947">T</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id61949">
																	<m:csymbol id="id61950" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id61955">g</m:ci>
																	<m:times id="id61957"/>
																</m:apply>
																<m:ci id="id61958">@</m:ci>
															</m:apply>
															<m:apply id="id61960">
																<m:times id="id61961"/>
																<m:apply id="id61962">
																	<m:csymbol id="id61964" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id61968">d</m:ci>
																	<m:ci id="id61970">p</m:ci>
																</m:apply>
																<m:apply id="id61972">
																	<m:csymbol id="id61974" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id61978">t</m:ci>
																	<m:times id="id61980"/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id61982" class="eqpad"/>
									</tr>

								</table>
								<p id="id61989" style="" class="p">where the projection <m:math id="id61995" display="inline">
										<m:semantics id="id61999">
											<m:mi id="id62000">p</m:mi>
											<m:annotation-xml id="id62002" encoding="MathML-Content">
												<m:ci id="id62005">p</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is induced by the inclusion <m:math id="id62009" display="inline">
										<m:semantics id="id62012">
											<m:mrow id="id62013">
												<m:mi id="id62014">T</m:mi>
												<m:mo id="id62016">↪</m:mo>
												<m:mi id="id62019">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id62021" encoding="MathML-Content">
												<m:apply id="id62024">
													<m:ci id="id62025">↪</m:ci>
													<m:ci id="id62028">T</m:ci>
													<m:ci id="id62030">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
Define <m:math id="id62033" display="inline">
										<m:semantics id="id62037">
											<m:mrow id="id62038">
												<m:mi id="id62039">V</m:mi>
												<m:mo id="id62041">⊂</m:mo>
												<m:msup id="id62043">
													<m:mi id="id62044" mathvariant="fraktur">g</m:mi>
													<m:mo id="id62049">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id62051" encoding="MathML-Content">
												<m:apply id="id62054">
													<m:ci id="id62055">⊂</m:ci>
													<m:ci id="id62058">V</m:ci>
													<m:apply id="id62060">
														<m:csymbol id="id62061" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id62066">g</m:ci>
														<m:times id="id62068"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> by <m:math id="id62070" display="inline">
										<m:semantics id="id62074">
											<m:mrow id="id62075">
												<m:mi id="id62076">V</m:mi>
												<m:mo id="id62078">:=</m:mo>
												<m:mrow id="id62080">
													<m:msup id="id62081">
														<m:mi id="id62082">p</m:mi>
														<m:mrow id="id62084">
															<m:mo id="id62085">-</m:mo>
															<m:mn id="id62087">1</m:mn>
														</m:mrow>
													</m:msup>
													<m:mo id="id62090">⁢</m:mo>
													<m:mfenced id="id62092" open="(" close=")">
														<m:mn id="id62097">0</m:mn>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id62099" encoding="MathML-Content">
												<m:apply id="id62102">
													<m:ci id="id62104">:=</m:ci>
													<m:ci id="id62106">V</m:ci>
													<m:apply id="id62108">
														<m:times id="id62109"/>
														<m:apply id="id62110">
															<m:csymbol id="id62111" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id62116">p</m:ci>
															<m:apply id="id62118">
																<m:minus id="id62119"/>
																<m:cn id="id62120">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id62122">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Then <m:math id="id62126" display="inline">
										<m:semantics id="id62129">
											<m:mrow id="id62130">
												<m:mrow id="id62131">
													<m:msubsup id="id62132">
														<m:mi id="id62133">μ</m:mi>
														<m:mi id="id62136">T</m:mi>
														<m:mrow id="id62138">
															<m:mo id="id62139">-</m:mo>
															<m:mn id="id62141">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id62143">⁢</m:mo>
													<m:mfenced id="id62145" open="(" close=")">
														<m:mn id="id62150">0</m:mn>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id62153">=</m:mo>
												<m:mrow id="id62155">
													<m:msubsup id="id62156">
														<m:mi id="id62157">μ</m:mi>
														<m:mi id="id62159">G</m:mi>
														<m:mrow id="id62161">
															<m:mo id="id62162">-</m:mo>
															<m:mn id="id62165">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id62167">⁢</m:mo>
													<m:mfenced id="id62169" open="(" close=")">
														<m:mi id="id62174">V</m:mi>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id62176" encoding="MathML-Content">
												<m:apply id="id62180">
													<m:eq id="id62181"/>
													<m:apply id="id62182">
														<m:times id="id62183"/>
														<m:apply id="id62184">
															<m:csymbol id="id62185" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id62190">
																<m:csymbol id="id62191" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id62195">μ</m:ci>
																<m:ci id="id62198">T</m:ci>
															</m:apply>
															<m:apply id="id62200">
																<m:minus id="id62201"/>
																<m:cn id="id62202">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id62204">0</m:cn>
													</m:apply>
													<m:apply id="id62206">
														<m:times id="id62207"/>
														<m:apply id="id62208">
															<m:csymbol id="id62210" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id62214">
																<m:csymbol id="id62215" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id62220">μ</m:ci>
																<m:ci id="id62222">G</m:ci>
															</m:apply>
															<m:apply id="id62224">
																<m:minus id="id62226"/>
																<m:cn id="id62227">1</m:cn>
															</m:apply>
														</m:apply>
														<m:ci id="id62229">V</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the
fact that <m:math id="id62232" display="inline">
										<m:semantics id="id62236">
											<m:mrow id="id62237">
												<m:mn id="id62238">0</m:mn>
												<m:mo id="id62240">∈</m:mo>
												<m:msup id="id62242">
													<m:mi id="id62243" mathvariant="fraktur">t</m:mi>
													<m:mo id="id62248">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id62250" encoding="MathML-Content">
												<m:apply id="id62253">
													<m:ci id="id62254">∈</m:ci>
													<m:cn id="id62257">0</m:cn>
													<m:apply id="id62259">
														<m:csymbol id="id62260" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id62264">t</m:ci>
														<m:times id="id62267"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a regular value for <m:math id="id62269" display="inline">
										<m:semantics id="id62272">
											<m:msub id="id62274">
												<m:mi id="id62275">μ</m:mi>
												<m:mi id="id62277">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id62279" encoding="MathML-Content">
												<m:apply id="id62282">
													<m:csymbol id="id62284" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id62288">μ</m:ci>
													<m:ci id="id62291">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is equivalent to the
assertion that the subspace <m:math id="id62294" display="inline">
										<m:semantics id="id62298">
											<m:mi id="id62299">V</m:mi>
											<m:annotation-xml id="id62301" encoding="MathML-Content">
												<m:ci id="id62304">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is transverse to the map <m:math id="id62308" display="inline">
										<m:semantics id="id62311">
											<m:msub id="id62312">
												<m:mi id="id62313">μ</m:mi>
												<m:mi id="id62316">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id62318" encoding="MathML-Content">
												<m:apply id="id62321">
													<m:csymbol id="id62322" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id62327">μ</m:ci>
													<m:ci id="id62329">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Note
that <m:math id="id62333" display="inline">
										<m:semantics id="id62336">
											<m:msub id="id62337">
												<m:mi id="id62338">μ</m:mi>
												<m:mi id="id62341">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id62343" encoding="MathML-Content">
												<m:apply id="id62346">
													<m:csymbol id="id62347" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id62352">μ</m:ci>
													<m:ci id="id62354">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a <m:math id="id62358" display="inline">
										<m:semantics id="id62361">
											<m:mi id="id62362">T</m:mi>
											<m:annotation-xml id="id62364" encoding="MathML-Content">
												<m:ci id="id62368">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant map and the coadjoint action of <m:math id="id62371" display="inline">
										<m:semantics id="id62375">
											<m:mi id="id62376">T</m:mi>
											<m:annotation-xml id="id62378" encoding="MathML-Content">
												<m:ci id="id62381">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
on <m:math id="id62385" display="inline">
										<m:semantics id="id62388">
											<m:msup id="id62389">
												<m:mi id="id62390" mathvariant="fraktur">g</m:mi>
												<m:mo id="id62394">*</m:mo>
											</m:msup>
											<m:annotation-xml id="id62397" encoding="MathML-Content">
												<m:apply id="id62400">
													<m:csymbol id="id62401" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id62406">g</m:ci>
													<m:times id="id62408"/>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> preserves the subspace <m:math id="id62410" display="inline">
										<m:semantics id="id62414">
											<m:mi id="id62415">V</m:mi>
											<m:annotation-xml id="id62417" encoding="MathML-Content">
												<m:ci id="id62420">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Moreover, given our choice of
positive and negative roots, we have
									<m:math id="id62424" display="inline">
										<m:semantics id="id62427">
											<m:mrow id="id62428">
												<m:mi id="id62430">V</m:mi>
												<m:mo id="id62432">≅</m:mo>
												<m:mrow id="id62434">
													<m:msub id="id62435">
														<m:mo id="id62436">⊕</m:mo>
														<m:mrow id="id62439">
															<m:mi id="id62440">α</m:mi>
															<m:mo id="id62442">∈</m:mo>
															<m:msup id="id62444">
																<m:mi id="id62446" mathvariant="normal">Δ</m:mi>
																<m:mo id="id62450">-</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:msub id="id62452">
														<m:mi id="id62453" mathvariant="double-struck">C</m:mi>
														<m:mfenced id="id62458" open="(" close=")">
															<m:mi id="id62463">α</m:mi>
														</m:mfenced>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id62465" encoding="MathML-Content">
												<m:apply id="id62469">
													<m:approx id="id62470"/>
													<m:ci id="id62471">V</m:ci>
													<m:apply id="id62473">
														<m:apply id="id62474">
															<m:csymbol id="id62475" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id62480">⊕</m:ci>
															<m:apply id="id62482">
																<m:ci id="id62483">∈</m:ci>
																<m:ci id="id62486">α</m:ci>
																<m:apply id="id62488">
																	<m:csymbol id="id62489" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id62494">Δ</m:ci>
																	<m:minus id="id62496"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id62497">
															<m:csymbol id="id62498" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id62503">C</m:ci>
															<m:ci id="id62505">α</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p7">
								<p id="id62517" style="" class="p">The restriction of <m:math id="id62524" display="inline">
										<m:semantics id="id62527">
											<m:msub id="id62528">
												<m:mi id="id62529">μ</m:mi>
												<m:mi id="id62532">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id62534" encoding="MathML-Content">
												<m:apply id="id62537">
													<m:csymbol id="id62538" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id62543">μ</m:ci>
													<m:ci id="id62545">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to <m:math id="id62549" display="inline">
										<m:semantics id="id62552">
											<m:mrow id="id62553">
												<m:msubsup id="id62554">
													<m:mi id="id62555">μ</m:mi>
													<m:mi id="id62558">T</m:mi>
													<m:mrow id="id62560">
														<m:mo id="id62561">-</m:mo>
														<m:mn id="id62563">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id62565">⁢</m:mo>
												<m:mfenced id="id62567" open="(" close=")">
													<m:mn id="id62572">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id62575" encoding="MathML-Content">
												<m:apply id="id62578">
													<m:times id="id62579"/>
													<m:apply id="id62580">
														<m:csymbol id="id62581" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id62586">
															<m:csymbol id="id62587" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id62592">μ</m:ci>
															<m:ci id="id62594">T</m:ci>
														</m:apply>
														<m:apply id="id62596">
															<m:minus id="id62597"/>
															<m:cn id="id62598">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id62600">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> defines a <m:math id="id62604" display="inline">
										<m:semantics id="id62607">
											<m:mi id="id62608">T</m:mi>
											<m:annotation-xml id="id62610" encoding="MathML-Content">
												<m:ci id="id62614">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant map
									<m:math id="id62617" display="inline">
										<m:semantics id="id62620">
											<m:mrow id="id62622">
												<m:mover id="id62623" accent="true">
													<m:mi id="id62626">s</m:mi>
													<m:mo id="id62628">~</m:mo>
												</m:mover>
												<m:mo id="id62630">:</m:mo>
												<m:mrow id="id62632">
													<m:mrow id="id62633">
														<m:msubsup id="id62634">
															<m:mi id="id62636">μ</m:mi>
															<m:mi id="id62638">T</m:mi>
															<m:mrow id="id62640">
																<m:mo id="id62641">-</m:mo>
																<m:mn id="id62643">1</m:mn>
															</m:mrow>
														</m:msubsup>
														<m:mo id="id62645">⁢</m:mo>
														<m:mfenced id="id62648" open="(" close=")">
															<m:mn id="id62653">0</m:mn>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id62655">→</m:mo>
													<m:mi id="id62657">V</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id62660" encoding="MathML-Content">
												<m:apply id="id62663">
													<m:ci id="id62664">:</m:ci>
													<m:apply id="id62666">
														<m:ci id="id62667">~</m:ci>
														<m:ci id="id62669">s</m:ci>
													</m:apply>
													<m:apply id="id62671">
														<m:ci id="id62672">→</m:ci>
														<m:apply id="id62675">
															<m:times id="id62676"/>
															<m:apply id="id62677">
																<m:csymbol id="id62678" cd="ambiguous">superscript</m:csymbol>
																<m:apply id="id62683">
																	<m:csymbol id="id62684" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id62688">μ</m:ci>
																	<m:ci id="id62691">T</m:ci>
																</m:apply>
																<m:apply id="id62693">
																	<m:minus id="id62694"/>
																	<m:cn id="id62695">1</m:cn>
																</m:apply>
															</m:apply>
															<m:cn id="id62697">0</m:cn>
														</m:apply>
														<m:ci id="id62699">V</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and taking the quotient by <m:math id="id62703" display="inline">
										<m:semantics id="id62706">
											<m:mi id="id62707">T</m:mi>
											<m:annotation-xml id="id62710" encoding="MathML-Content">
												<m:ci id="id62713">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, then <m:math id="id62716" display="inline">
										<m:semantics id="id62720">
											<m:mover id="id62721" accent="true">
												<m:mi id="id62724">s</m:mi>
												<m:mo id="id62726">~</m:mo>
											</m:mover>
											<m:annotation-xml id="id62728" encoding="MathML-Content">
												<m:apply id="id62732">
													<m:ci id="id62733">~</m:ci>
													<m:ci id="id62735">s</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
defines a section <m:math id="id62738" display="inline">
										<m:semantics id="id62742">
											<m:mi id="id62743">s</m:mi>
											<m:annotation-xml id="id62745" encoding="MathML-Content">
												<m:ci id="id62748">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of the associated bundle <m:math id="id62752" display="inline">
										<m:semantics id="id62755">
											<m:mrow id="id62756">
												<m:msubsup id="id62757">
													<m:mi id="id62758">μ</m:mi>
													<m:mi id="id62761">T</m:mi>
													<m:mrow id="id62763">
														<m:mo id="id62764">-</m:mo>
														<m:mn id="id62766">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id62768">(</m:mo>
												<m:mn id="id62770">0</m:mn>
												<m:mo id="id62772">)</m:mo>
												<m:msub id="id62775">
													<m:mo id="id62776">×</m:mo>
													<m:mi id="id62778">T</m:mi>
												</m:msub>
												<m:mi id="id62780">V</m:mi>
												<m:mo id="id62782">→</m:mo>
												<m:mi id="id62785">X</m:mi>
												<m:mo id="id62787">/</m:mo>
												<m:mo id="id62789">/</m:mo>
												<m:mi id="id62791">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id62793" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>.
Since <m:math id="id62798" display="inline">
										<m:semantics id="id62801">
											<m:mrow id="id62802">
												<m:mn id="id62803">0</m:mn>
												<m:mo id="id62806">∈</m:mo>
												<m:msup id="id62808">
													<m:mi id="id62809" mathvariant="fraktur">g</m:mi>
													<m:mo id="id62813">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id62816" encoding="MathML-Content">
												<m:apply id="id62819">
													<m:ci id="id62820">∈</m:ci>
													<m:cn id="id62822">0</m:cn>
													<m:apply id="id62824">
														<m:csymbol id="id62826" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id62830">g</m:ci>
														<m:times id="id62832"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a regular value of <m:math id="id62835" display="inline">
										<m:semantics id="id62838">
											<m:msub id="id62839">
												<m:mi id="id62840">μ</m:mi>
												<m:mi id="id62843">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id62845" encoding="MathML-Content">
												<m:apply id="id62848">
													<m:csymbol id="id62849" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id62854">μ</m:ci>
													<m:ci id="id62856">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> it follows that <m:math id="id62860" display="inline">
										<m:semantics id="id62863">
											<m:mrow id="id62864">
												<m:mn id="id62865">0</m:mn>
												<m:mo id="id62867">∈</m:mo>
												<m:mi id="id62870">V</m:mi>
											</m:mrow>
											<m:annotation-xml id="id62872" encoding="MathML-Content">
												<m:apply id="id62875">
													<m:ci id="id62876">∈</m:ci>
													<m:cn id="id62879">0</m:cn>
													<m:ci id="id62881">V</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
is a regular value of <m:math id="id62884" display="inline">
										<m:semantics id="id62888">
											<m:mover id="id62889" accent="true">
												<m:mi id="id62892">s</m:mi>
												<m:mo id="id62894">~</m:mo>
											</m:mover>
											<m:annotation-xml id="id62896" encoding="MathML-Content">
												<m:apply id="id62900">
													<m:ci id="id62901">~</m:ci>
													<m:ci id="id62903">s</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and hence <m:math id="id62906" display="inline">
										<m:semantics id="id62910">
											<m:mi id="id62911">s</m:mi>
											<m:annotation-xml id="id62913" encoding="MathML-Content">
												<m:ci id="id62916">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is transverse to the
zero section.</p>
							</div>

							<div class="para" id="S1.p8">
								<p id="id62928" style="" class="p">Finally, the identification of <m:math id="id62935" display="inline">
										<m:semantics id="id62938">
											<m:mi id="id62939">V</m:mi>
											<m:annotation-xml id="id62942" encoding="MathML-Content">
												<m:ci id="id62945">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in terms of negative roots gives
									<m:math id="id62948" display="inline">
										<m:semantics id="id62952">
											<m:mrow id="id62953">
												<m:msubsup id="id62954">
													<m:mi id="id62955">μ</m:mi>
													<m:mi id="id62957">T</m:mi>
													<m:mrow id="id62960">
														<m:mo id="id62961">-</m:mo>
														<m:mn id="id62963">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id62965">(</m:mo>
												<m:mn id="id62967">0</m:mn>
												<m:mo id="id62969">)</m:mo>
												<m:msub id="id62971">
													<m:mo id="id62972">×</m:mo>
													<m:mi id="id62975">T</m:mi>
												</m:msub>
												<m:mi id="id62977">V</m:mi>
												<m:mo id="id62979">≅</m:mo>
												<m:msub id="id62981">
													<m:mo id="id62982">⊕</m:mo>
													<m:mrow id="id62985">
														<m:mi id="id62986">α</m:mi>
														<m:mo id="id62988">∈</m:mo>
														<m:msup id="id62991">
															<m:mi id="id62992" mathvariant="normal">Δ</m:mi>
															<m:mo id="id62996">-</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:msub id="id62999">
													<m:mi id="id63000">L</m:mi>
													<m:mi id="id63002">α</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id63004" encoding="MathML-Content">
												<m:apply id="id63008">
													<m:csymbol id="id63009" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id63013">
														<m:csymbol id="id63014" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id63019">μ</m:ci>
														<m:ci id="id63021">T</m:ci>
													</m:apply>
													<m:apply id="id63024">
														<m:minus id="id63025"/>
														<m:cn id="id63026">1</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p9">
								<p id="id63038" style="" class="p">
									<span id="id63043" style="" class="text italic">2. The fibering.</span>
We write <m:math id="id63051" display="inline">
										<m:semantics id="id63054">
											<m:mrow id="id63055">
												<m:mi id="id63056">Z</m:mi>
												<m:mo id="id63058">:=</m:mo>
												<m:mrow id="id63060">
													<m:msubsup id="id63062">
														<m:mi id="id63063">μ</m:mi>
														<m:mi id="id63065">G</m:mi>
														<m:mrow id="id63067">
															<m:mo id="id63068">-</m:mo>
															<m:mn id="id63070">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id63072">⁢</m:mo>
													<m:mfenced id="id63075" open="(" close=")">
														<m:mn id="id63080">0</m:mn>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id63082" encoding="MathML-Content">
												<m:apply id="id63085">
													<m:ci id="id63086">:=</m:ci>
													<m:ci id="id63089">Z</m:ci>
													<m:apply id="id63091">
														<m:times id="id63092"/>
														<m:apply id="id63093">
															<m:csymbol id="id63094" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id63099">
																<m:csymbol id="id63100" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id63104">μ</m:ci>
																<m:ci id="id63107">G</m:ci>
															</m:apply>
															<m:apply id="id63109">
																<m:minus id="id63110"/>
																<m:cn id="id63111">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id63113">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Let <m:math id="id63117" display="inline">
										<m:semantics id="id63120">
											<m:msub id="id63121">
												<m:mi id="id63122">π</m:mi>
												<m:mi id="id63124">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id63127" encoding="MathML-Content">
												<m:apply id="id63130">
													<m:csymbol id="id63131" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id63136">π</m:ci>
													<m:ci id="id63138">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id63142" display="inline">
										<m:semantics id="id63145">
											<m:msub id="id63146">
												<m:mi id="id63147">π</m:mi>
												<m:mi id="id63149">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id63152" encoding="MathML-Content">
												<m:apply id="id63155">
													<m:csymbol id="id63156" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id63161">π</m:ci>
													<m:ci id="id63163">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> denote the projections</p>
								<table class="equation" id="S1.Ex6">

									<tr valign="baseline" class="equation" id="S1.Ex6">
										<td id="id63183" class="eqpad"/>
										<td id="id63188" nowrap="yes" align="center" colspan="1">
											<m:math id="id63195" display="block">
												<m:semantics id="id63198">
													<m:mrow id="id63199">
														<m:mo id="id63200">*</m:mo>
														<m:mo id="id63202">*</m:mo>
														<m:mo id="id63205">[</m:mo>
														<m:mi id="id63207">l</m:mi>
														<m:mo id="id63209">]</m:mo>
														<m:msubsup id="id63211">
															<m:mi id="id63212">μ</m:mi>
															<m:mi id="id63214">G</m:mi>
															<m:mrow id="id63217">
																<m:mo id="id63218">-</m:mo>
																<m:mn id="id63220">1</m:mn>
															</m:mrow>
														</m:msubsup>
														<m:mo id="id63222">(</m:mo>
														<m:mn id="id63224">0</m:mn>
														<m:mo id="id63226">)</m:mo>
														<m:mo id="id63228">=</m:mo>
														<m:mo id="id63230">:</m:mo>
														<m:mi id="id63233">Z</m:mi>
														<m:mo id="id63235">[</m:mo>
														<m:mi id="id63237">r</m:mi>
														<m:msup id="id63239">
															<m:mo id="id63240">]</m:mo>
															<m:msub id="id63242">
																<m:mi id="id63243">π</m:mi>
																<m:mi id="id63246">T</m:mi>
															</m:msub>
														</m:msup>
														<m:mo id="id63248">[</m:mo>
														<m:mi id="id63250">r</m:mi>
														<m:mi id="id63252">d</m:mi>
														<m:msub id="id63254">
															<m:mo id="id63255">]</m:mo>
															<m:msub id="id63257">
																<m:mi id="id63258">π</m:mi>
																<m:mi id="id63261">G</m:mi>
															</m:msub>
														</m:msub>
														<m:mi id="id63263">Z</m:mi>
														<m:mo id="id63265">/</m:mo>
														<m:mi id="id63267">T</m:mi>
														<m:mo id="id63269">[</m:mo>
														<m:mi id="id63272">d</m:mi>
														<m:msup id="id63274">
															<m:mo id="id63275">]</m:mo>
															<m:mi id="id63277">π</m:mi>
														</m:msup>
														<m:mi id="id63279">Z</m:mi>
														<m:mo id="id63281">/</m:mo>
														<m:mi id="id63284">G</m:mi>
													</m:mrow>
													<m:annotation-xml id="id63286" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id63290" class="eqpad"/>
									</tr>

								</table>
								<p id="id63297" style="" class="p">(<m:math id="id63303" display="inline">
										<m:semantics id="id63306">
											<m:mi id="id63307">π</m:mi>
											<m:annotation-xml id="id63310" encoding="MathML-Content">
												<m:ci id="id63313">π</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> was defined above the proposition).</p>
							</div>

							<div class="para" id="S1.p10">
								<p id="id63326" style="" class="p">Consider the foliation of <m:math id="id63332" display="inline">
										<m:semantics id="id63335">
											<m:mi id="id63336">Z</m:mi>
											<m:annotation-xml id="id63339" encoding="MathML-Content">
												<m:ci id="id63342">Z</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> given by the <m:math id="id63345" display="inline">
										<m:semantics id="id63349">
											<m:mi id="id63350">G</m:mi>
											<m:annotation-xml id="id63352" encoding="MathML-Content">
												<m:ci id="id63355">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-orbits: using a
									<m:math id="id63359" display="inline">
										<m:semantics id="id63362">
											<m:mi id="id63363">G</m:mi>
											<m:annotation-xml id="id63365" encoding="MathML-Content">
												<m:ci id="id63369">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-invariant metric to take orthgonal complements, and the
infinitesimal action to identify tangents to the <m:math id="id63373" display="inline">
										<m:semantics id="id63376">
											<m:mi id="id63377">G</m:mi>
											<m:annotation-xml id="id63379" encoding="MathML-Content">
												<m:ci id="id63383">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-orbits, we have
the <m:math id="id63386" display="inline">
										<m:semantics id="id63390">
											<m:mi id="id63391">G</m:mi>
											<m:annotation-xml id="id63393" encoding="MathML-Content">
												<m:ci id="id63396">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant identification</p>
								<table class="equation" id="S1.Ex7">

									<tr valign="baseline" class="equation" id="S1.Ex7">
										<td id="id63416" class="eqpad"/>
										<td id="id63421" nowrap="yes" align="center" colspan="1">
											<m:math id="id63428" display="block">
												<m:semantics id="id63431">
													<m:mrow id="id63432">
														<m:mrow id="id63434">
															<m:mrow id="id63435">
																<m:mi id="id63436">T</m:mi>
																<m:mo id="id63438">⁢</m:mo>
																<m:mi id="id63440">Z</m:mi>
															</m:mrow>
															<m:mo id="id63442">≅</m:mo>
															<m:mrow id="id63445">
																<m:mfenced id="id63446" open="(" close=")">
																	<m:mrow id="id63451">
																		<m:mi id="id63452">Z</m:mi>
																		<m:mo id="id63454">×</m:mo>
																		<m:mi id="id63456" mathvariant="fraktur">g</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo id="id63461">⊕</m:mo>
																<m:mrow id="id63463">
																	<m:msubsup id="id63464">
																		<m:mi id="id63465">π</m:mi>
																		<m:mi id="id63468">G</m:mi>
																		<m:mo id="id63470">*</m:mo>
																	</m:msubsup>
																	<m:mo id="id63472">⁢</m:mo>
																	<m:mi id="id63474">T</m:mi>
																	<m:mo id="id63477">⁢</m:mo>
																	<m:mfenced id="id63479" open="(" close=")">
																		<m:mrow id="id63484">
																			<m:mi id="id63485">Z</m:mi>
																			<m:mo id="id63487">/</m:mo>
																			<m:mi id="id63489">G</m:mi>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:mo id="id63492">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id63494" encoding="MathML-Content">
														<m:apply id="id63497">
															<m:approx id="id63498"/>
															<m:apply id="id63499">
																<m:times id="id63500"/>
																<m:ci id="id63501">T</m:ci>
																<m:ci id="id63503">Z</m:ci>
															</m:apply>
															<m:apply id="id63506">
																<m:ci id="id63507">⊕</m:ci>
																<m:apply id="id63509">
																	<m:times id="id63510"/>
																	<m:ci id="id63511">Z</m:ci>
																	<m:ci id="id63513">g</m:ci>
																</m:apply>
																<m:apply id="id63515">
																	<m:times id="id63516"/>
																	<m:apply id="id63518">
																		<m:csymbol id="id63519" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id63523">
																			<m:csymbol id="id63524" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id63529">π</m:ci>
																			<m:ci id="id63531">G</m:ci>
																		</m:apply>
																		<m:times id="id63534"/>
																	</m:apply>
																	<m:ci id="id63535">T</m:ci>
																	<m:apply id="id63537">
																		<m:divide id="id63538"/>
																		<m:ci id="id63539">Z</m:ci>
																		<m:ci id="id63541">G</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id63544" class="eqpad"/>
									</tr>

								</table>
								<p id="id63551" style="" class="p">where <m:math id="id63557" display="inline">
										<m:semantics id="id63560">
											<m:mi id="id63562">G</m:mi>
											<m:annotation-xml id="id63564" encoding="MathML-Content">
												<m:ci id="id63567">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> acts on <m:math id="id63570" display="inline">
										<m:semantics id="id63574">
											<m:mi id="id63575" mathvariant="fraktur">g</m:mi>
											<m:annotation-xml id="id63579" encoding="MathML-Content">
												<m:ci id="id63583">g</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> by the adjoint action.
Restricting to the action of <m:math id="id63586" display="inline">
										<m:semantics id="id63590">
											<m:mi id="id63591">T</m:mi>
											<m:annotation-xml id="id63593" encoding="MathML-Content">
												<m:ci id="id63596">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, we can refine this identification
using the <m:math id="id63600" display="inline">
										<m:semantics id="id63603">
											<m:mi id="id63604">T</m:mi>
											<m:annotation-xml id="id63606" encoding="MathML-Content">
												<m:ci id="id63610">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant decomposition <m:math id="id63613" display="inline">
										<m:semantics id="id63617">
											<m:mrow id="id63618">
												<m:mi id="id63619" mathvariant="fraktur">g</m:mi>
												<m:mo id="id63623">≅</m:mo>
												<m:mrow id="id63626">
													<m:mi id="id63627" mathvariant="fraktur">t</m:mi>
													<m:mo id="id63631">⊕</m:mo>
													<m:mi id="id63634" mathvariant="fraktur">v</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id63638" encoding="MathML-Content">
												<m:apply id="id63641">
													<m:approx id="id63642"/>
													<m:ci id="id63643">g</m:ci>
													<m:apply id="id63646">
														<m:ci id="id63647">⊕</m:ci>
														<m:ci id="id63649">t</m:ci>
														<m:ci id="id63651">v</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
								</p>
								<table class="equation" id="S1.Ex8">

									<tr valign="baseline" class="equation" id="S1.Ex8">
										<td id="id63670" class="eqpad"/>
										<td id="id63674" nowrap="yes" align="center" colspan="1">
											<m:math id="id63682" display="block">
												<m:semantics id="id63685">
													<m:mrow id="id63686">
														<m:mrow id="id63687">
															<m:mrow id="id63688">
																<m:mi id="id63689">T</m:mi>
																<m:mo id="id63691">⁢</m:mo>
																<m:mi id="id63694">Z</m:mi>
															</m:mrow>
															<m:mo id="id63696">≅</m:mo>
															<m:mrow id="id63698">
																<m:mfenced id="id63699" open="(" close=")">
																	<m:mrow id="id63704">
																		<m:mi id="id63706">Z</m:mi>
																		<m:mo id="id63708">×</m:mo>
																		<m:mi id="id63710" mathvariant="fraktur">t</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo id="id63714">⊕</m:mo>
																<m:mfenced id="id63717" open="(" close=")">
																	<m:mrow id="id63722">
																		<m:mi id="id63723">Z</m:mi>
																		<m:mo id="id63725">×</m:mo>
																		<m:mi id="id63728" mathvariant="fraktur">v</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo id="id63732">⊕</m:mo>
																<m:mrow id="id63734">
																	<m:msubsup id="id63735">
																		<m:mi id="id63736">π</m:mi>
																		<m:mi id="id63739">G</m:mi>
																		<m:mo id="id63741">*</m:mo>
																	</m:msubsup>
																	<m:mo id="id63743">⁢</m:mo>
																	<m:mi id="id63746">T</m:mi>
																	<m:mo id="id63748">⁢</m:mo>
																	<m:mfenced id="id63750" open="(" close=")">
																		<m:mrow id="id63755">
																			<m:mi id="id63756">Z</m:mi>
																			<m:mo id="id63758">/</m:mo>
																			<m:mi id="id63760">G</m:mi>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:mo id="id63763">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id63765" encoding="MathML-Content">
														<m:apply id="id63768">
															<m:approx id="id63769"/>
															<m:apply id="id63770">
																<m:times id="id63771"/>
																<m:ci id="id63772">T</m:ci>
																<m:ci id="id63774">Z</m:ci>
															</m:apply>
															<m:apply id="id63777">
																<m:ci id="id63778">⊕</m:ci>
																<m:apply id="id63780">
																	<m:times id="id63781"/>
																	<m:ci id="id63782">Z</m:ci>
																	<m:ci id="id63784">t</m:ci>
																</m:apply>
																<m:apply id="id63786">
																	<m:times id="id63788"/>
																	<m:ci id="id63789">Z</m:ci>
																	<m:ci id="id63791">v</m:ci>
																</m:apply>
																<m:apply id="id63793">
																	<m:times id="id63794"/>
																	<m:apply id="id63795">
																		<m:csymbol id="id63796" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id63801">
																			<m:csymbol id="id63802" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id63806">π</m:ci>
																			<m:ci id="id63809">G</m:ci>
																		</m:apply>
																		<m:times id="id63811"/>
																	</m:apply>
																	<m:ci id="id63812">T</m:ci>
																	<m:apply id="id63814">
																		<m:divide id="id63815"/>
																		<m:ci id="id63816">Z</m:ci>
																		<m:ci id="id63818">G</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id63822" class="eqpad"/>
									</tr>

								</table>
								<p id="id63828" style="" class="p">with <m:math id="id63835" display="inline">
										<m:semantics id="id63838">
											<m:mrow id="id63839">
												<m:mi id="id63840" mathvariant="fraktur">v</m:mi>
												<m:mo id="id63844">≅</m:mo>
												<m:mrow id="id63847">
													<m:msub id="id63848">
														<m:mo id="id63849">⊕</m:mo>
														<m:mrow id="id63851">
															<m:mi id="id63852">α</m:mi>
															<m:mo id="id63855">∈</m:mo>
															<m:msup id="id63857">
																<m:mi id="id63858" mathvariant="normal">Δ</m:mi>
																<m:mo id="id63863">+</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:msub id="id63865">
														<m:mi id="id63866" mathvariant="double-struck">C</m:mi>
														<m:mfenced id="id63871" open="(" close=")">
															<m:mi id="id63876">α</m:mi>
														</m:mfenced>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id63878" encoding="MathML-Content">
												<m:apply id="id63881">
													<m:approx id="id63882"/>
													<m:ci id="id63884">v</m:ci>
													<m:apply id="id63886">
														<m:apply id="id63887">
															<m:csymbol id="id63888" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id63892">⊕</m:ci>
															<m:apply id="id63895">
																<m:ci id="id63896">∈</m:ci>
																<m:ci id="id63898">α</m:ci>
																<m:apply id="id63901">
																	<m:csymbol id="id63902" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id63906">Δ</m:ci>
																	<m:plus id="id63909"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id63910">
															<m:csymbol id="id63911" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id63916">C</m:ci>
															<m:ci id="id63918">α</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
Identifying the <m:math id="id63922" display="inline">
										<m:semantics id="id63925">
											<m:mrow id="id63926">
												<m:mi id="id63927">Z</m:mi>
												<m:mo id="id63929">×</m:mo>
												<m:mi id="id63932" mathvariant="fraktur">t</m:mi>
											</m:mrow>
											<m:annotation-xml id="id63936" encoding="MathML-Content">
												<m:apply id="id63939">
													<m:times id="id63940"/>
													<m:ci id="id63941">Z</m:ci>
													<m:ci id="id63944">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> factor as the tangents to the <m:math id="id63947" display="inline">
										<m:semantics id="id63950">
											<m:mi id="id63952">T</m:mi>
											<m:annotation-xml id="id63954" encoding="MathML-Content">
												<m:ci id="id63957">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-orbits,
and taking the quotient by <m:math id="id63961" display="inline">
										<m:semantics id="id63964">
											<m:mi id="id63965">T</m:mi>
											<m:annotation-xml id="id63967" encoding="MathML-Content">
												<m:ci id="id63970">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>,</p>
								<table class="equation" id="S1.Ex9">

									<tr valign="baseline" class="equation" id="S1.Ex9">
										<td id="id63990" class="eqpad"/>
										<td id="id63995" nowrap="yes" align="center" colspan="1">
											<m:math id="id64002" display="block">
												<m:semantics id="id64005">
													<m:mrow id="id64006">
														<m:mi id="id64008">T</m:mi>
														<m:mo id="id64010">(</m:mo>
														<m:mi id="id64012">Z</m:mi>
														<m:mo id="id64014">/</m:mo>
														<m:mi id="id64016">T</m:mi>
														<m:mo id="id64018">)</m:mo>
														<m:mo id="id64020">≅</m:mo>
														<m:mo id="id64023">(</m:mo>
														<m:mi id="id64025">Z</m:mi>
														<m:msub id="id64027">
															<m:mo id="id64028">×</m:mo>
															<m:mi id="id64030">T</m:mi>
														</m:msub>
														<m:mi id="id64033" mathvariant="fraktur">v</m:mi>
														<m:mo id="id64037">)</m:mo>
														<m:mo id="id64039">⊕</m:mo>
														<m:msup id="id64042">
															<m:mi id="id64043">π</m:mi>
															<m:mo id="id64045">*</m:mo>
														</m:msup>
														<m:mi id="id64047">T</m:mi>
														<m:mo id="id64049">(</m:mo>
														<m:mi id="id64051">Z</m:mi>
														<m:mo id="id64054">/</m:mo>
														<m:mi id="id64056">G</m:mi>
														<m:mo id="id64058">)</m:mo>
														<m:mo id="id64060">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id64062" encoding="MathML-Content">
														<m:ci id="id64065">T</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id64069" class="eqpad"/>
									</tr>

								</table>
								<p id="id64075" style="" class="p">Hence, identifying <m:math id="id64082" display="inline">
										<m:semantics id="id64085">
											<m:mi id="id64086" mathvariant="fraktur">v</m:mi>
											<m:annotation-xml id="id64090" encoding="MathML-Content">
												<m:ci id="id64094">v</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in terms of positive roots gives
									<m:math id="id64097" display="inline">
										<m:semantics id="id64101">
											<m:mrow id="id64102">
												<m:mrow id="id64103">
													<m:mi id="id64104">v</m:mi>
													<m:mo id="id64106">⁢</m:mo>
													<m:mi id="id64108">e</m:mi>
													<m:mo id="id64110">⁢</m:mo>
													<m:mi id="id64113">r</m:mi>
													<m:mo id="id64115">⁢</m:mo>
													<m:mi id="id64117">t</m:mi>
												</m:mrow>
												<m:mo id="id64120">(</m:mo>
												<m:mi id="id64122">π</m:mi>
												<m:mo id="id64124">)</m:mo>
												<m:mo id="id64126">≅</m:mo>
												<m:mi id="id64129">Z</m:mi>
												<m:msub id="id64131">
													<m:mo id="id64132">×</m:mo>
													<m:mi id="id64134">T</m:mi>
												</m:msub>
												<m:mi id="id64136" mathvariant="fraktur">v</m:mi>
												<m:mo id="id64141">≅</m:mo>
												<m:msub id="id64143">
													<m:mo id="id64144">⊕</m:mo>
													<m:mrow id="id64147">
														<m:mi id="id64148">α</m:mi>
														<m:mo id="id64150">∈</m:mo>
														<m:msup id="id64152">
															<m:mi id="id64154" mathvariant="normal">Δ</m:mi>
															<m:mo id="id64158">+</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:mmultiscripts id="id64160">
													<m:mi id="id64161">L</m:mi>
													<m:mi id="id64164">α</m:mi>
													<m:none id="id64166"/>
													<m:mrow id="id64167">
														<m:mi id="id64168">Z</m:mi>
														<m:mo id="id64170">/</m:mo>
														<m:mi id="id64172">T</m:mi>
													</m:mrow>
													<m:none id="id64174"/>
												</m:mmultiscripts>
											</m:mrow>
											<m:annotation-xml id="id64176" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p11">
								<p id="id64189" style="" class="p">
									<span id="id64194" style="" class="text italic">3. The orientation.</span>
We begin by summarizing the arguments in the final stage of the proof.
We will first describe the symplectic form of
									<m:math id="id64203" display="inline">
										<m:semantics id="id64206">
											<m:mrow id="id64207">
												<m:mi id="id64208">X</m:mi>
												<m:mo id="id64210">/</m:mo>
												<m:mo id="id64212">/</m:mo>
												<m:mi id="id64214">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64217" encoding="MathML-Content">
												<m:ci id="id64220">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, restricted to <m:math id="id64223" display="inline">
										<m:semantics id="id64227">
											<m:mrow id="id64228">
												<m:mi id="id64229">Z</m:mi>
												<m:mo id="id64231">/</m:mo>
												<m:mi id="id64233">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64235" encoding="MathML-Content">
												<m:apply id="id64239">
													<m:divide id="id64240"/>
													<m:ci id="id64241">Z</m:ci>
													<m:ci id="id64243">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, in terms of a decomposition of
the tangent bundle (equation (<a id="id64249" href="#S1.E4" title="1.4 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id64247" style="" class="text tag">1.4</span>
									</a>) below).
We then describe an almost complex structure <m:math id="id64264" display="inline">
										<m:semantics id="id64267">
											<m:msub id="id64268">
												<m:mover id="id64269" accent="true">
													<m:mi id="id64272">J</m:mi>
													<m:mo id="id64275">~</m:mo>
												</m:mover>
												<m:mn id="id64277">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id64279" encoding="MathML-Content">
												<m:apply id="id64282">
													<m:csymbol id="id64283" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id64288">
														<m:ci id="id64289">~</m:ci>
														<m:ci id="id64291">J</m:ci>
													</m:apply>
													<m:cn id="id64293">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on <m:math id="id64297" display="inline">
										<m:semantics id="id64300">
											<m:mrow id="id64301">
												<m:mi id="id64302">X</m:mi>
												<m:mo id="id64304">/</m:mo>
												<m:mo id="id64306">/</m:mo>
												<m:mi id="id64309">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64311" encoding="MathML-Content">
												<m:ci id="id64314">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
which is compatible with the symplectic form on <m:math id="id64318" display="inline">
										<m:semantics id="id64321">
											<m:mrow id="id64322">
												<m:mi id="id64323">X</m:mi>
												<m:mo id="id64325">/</m:mo>
												<m:mo id="id64328">/</m:mo>
												<m:mi id="id64330">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64332" encoding="MathML-Content">
												<m:ci id="id64335">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and which
has a simple description over <m:math id="id64339" display="inline">
										<m:semantics id="id64342">
											<m:mrow id="id64343">
												<m:mi id="id64344">Z</m:mi>
												<m:mo id="id64346">/</m:mo>
												<m:mi id="id64349">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64351" encoding="MathML-Content">
												<m:apply id="id64354">
													<m:divide id="id64355"/>
													<m:ci id="id64356">Z</m:ci>
													<m:ci id="id64358">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
Finally, we show that <m:math id="id64362" display="inline">
										<m:semantics id="id64365">
											<m:msub id="id64366">
												<m:mover id="id64367" accent="true">
													<m:mi id="id64371">J</m:mi>
													<m:mo id="id64373">~</m:mo>
												</m:mover>
												<m:mn id="id64375">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id64377" encoding="MathML-Content">
												<m:apply id="id64380">
													<m:csymbol id="id64382" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id64386">
														<m:ci id="id64387">~</m:ci>
														<m:ci id="id64389">J</m:ci>
													</m:apply>
													<m:cn id="id64392">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is homotopic, through almost
complex structures, to an almost complex structure <m:math id="id64396" display="inline">
										<m:semantics id="id64399">
											<m:msub id="id64400">
												<m:mover id="id64401" accent="true">
													<m:mi id="id64404">J</m:mi>
													<m:mo id="id64406">~</m:mo>
												</m:mover>
												<m:mn id="id64409">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id64411" encoding="MathML-Content">
												<m:apply id="id64414">
													<m:csymbol id="id64415" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id64420">
														<m:ci id="id64421">~</m:ci>
														<m:ci id="id64423">J</m:ci>
													</m:apply>
													<m:cn id="id64425">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on
									<m:math id="id64429" display="inline">
										<m:semantics id="id64432">
											<m:mrow id="id64433">
												<m:mi id="id64434">X</m:mi>
												<m:mo id="id64436">/</m:mo>
												<m:mo id="id64438">/</m:mo>
												<m:mi id="id64440">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64443" encoding="MathML-Content">
												<m:ci id="id64446">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> with respect to which <m:math id="id64449" display="inline">
										<m:semantics id="id64453">
											<m:mrow id="id64454">
												<m:mi id="id64455">Z</m:mi>
												<m:mo id="id64457">/</m:mo>
												<m:mi id="id64459">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64461" encoding="MathML-Content">
												<m:apply id="id64465">
													<m:divide id="id64466"/>
													<m:ci id="id64467">Z</m:ci>
													<m:ci id="id64469">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is an almost complex submanifold,
and such that the almost complex structures on <m:math id="id64473" display="inline">
										<m:semantics id="id64476">
											<m:mrow id="id64477">
												<m:mi id="id64478">Z</m:mi>
												<m:mo id="id64480">/</m:mo>
												<m:mi id="id64483">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64485" encoding="MathML-Content">
												<m:apply id="id64488">
													<m:divide id="id64489"/>
													<m:ci id="id64490">Z</m:ci>
													<m:ci id="id64492">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and on its normal
bundle agree with the complex orientations described in the statement
of the proposition.</p>
							</div>

							<div class="para" id="S1.p12">
								<p id="id64505" style="" class="p">
									<span id="id64510" style="" class="text italic">3(i) The identification of the symplectic form.</span>
On <m:math id="id64519" display="inline">
										<m:semantics id="id64522">
											<m:mrow id="id64523">
												<m:mi id="id64524" mathvariant="fraktur">g</m:mi>
												<m:mo id="id64529">⊕</m:mo>
												<m:msup id="id64531">
													<m:mi id="id64532" mathvariant="fraktur">g</m:mi>
													<m:mo id="id64536">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id64539" encoding="MathML-Content">
												<m:apply id="id64542">
													<m:ci id="id64543">⊕</m:ci>
													<m:ci id="id64545">g</m:ci>
													<m:apply id="id64548">
														<m:csymbol id="id64549" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id64553">g</m:ci>
														<m:times id="id64555"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> we define the <m:math id="id64558" display="inline">
										<m:semantics id="id64561">
											<m:mi id="id64562">G</m:mi>
											<m:annotation-xml id="id64564" encoding="MathML-Content">
												<m:ci id="id64568">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-invariant symplectic form <m:math id="id64571" display="inline">
										<m:semantics id="id64575">
											<m:mi id="id64576">η</m:mi>
											<m:annotation-xml id="id64578" encoding="MathML-Content">
												<m:ci id="id64581">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
by using the duality pairing:
								</p>
								<table class="equation" id="S1.Ex10">

									<tr valign="baseline" class="equation" id="S1.Ex10">
										<td id="id64602" class="eqpad"/>
										<td id="id64606" nowrap="yes" align="center" colspan="1">
											<m:math id="id64614" display="block">
												<m:semantics id="id64617">
													<m:mrow id="id64618">
														<m:mrow id="id64619">
															<m:mrow id="id64620">
																<m:mrow id="id64621">
																	<m:mi id="id64622">η</m:mi>
																	<m:mo id="id64625">⁢</m:mo>
																	<m:mfenced id="id64627" open="(" close=")">
																		<m:mrow id="id64632">
																			<m:mi id="id64633">ξ</m:mi>
																			<m:mo id="id64636">,</m:mo>
																			<m:mi id="id64638">α</m:mi>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id64640">:=</m:mo>
																<m:mfenced id="id64644" open="⟨" close="⟩">
																	<m:mrow id="id64650">
																		<m:mi id="id64651">ξ</m:mi>
																		<m:mo id="id64653">,</m:mo>
																		<m:mi id="id64655">α</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id64658">,</m:mo>
															<m:mrow id="id64660">
																<m:mrow id="id64661">
																	<m:mrow id="id64662">
																		<m:mi id="id64663" mathvariant="normal">∀</m:mi>
																		<m:mi id="id64667">ξ</m:mi>
																	</m:mrow>
																	<m:mo id="id64669">∈</m:mo>
																	<m:mi id="id64672" mathvariant="fraktur">g</m:mi>
																</m:mrow>
																<m:mo id="id64675">,</m:mo>
																<m:mrow id="id64677">
																	<m:mi id="id64678">α</m:mi>
																	<m:mo id="id64681">∈</m:mo>
																	<m:msup id="id64683">
																		<m:mi id="id64684" mathvariant="fraktur">g</m:mi>
																		<m:mo id="id64689">*</m:mo>
																	</m:msup>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:mo id="id64691">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id64693" encoding="MathML-Content">
														<m:apply id="id64696">
															<m:ci id="id64697"/>
															<m:apply id="id64698">
																<m:ci id="id64699">:=</m:ci>
																<m:apply id="id64702">
																	<m:times id="id64703"/>
																	<m:ci id="id64704">η</m:ci>
																	<m:apply id="id64706">
																		<m:interval id="id64707" closure="open"/>
																		<m:ci id="id64710">ξ</m:ci>
																		<m:ci id="id64713">α</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id64715">
																	<m:list id="id64716"/>
																	<m:ci id="id64717">ξ</m:ci>
																	<m:ci id="id64720">α</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id64722">
																<m:ci id="id64723"/>
																<m:apply id="id64724">
																	<m:ci id="id64725">∈</m:ci>
																	<m:apply id="id64728">
																		<m:ci id="id64729">∀</m:ci>
																		<m:ci id="id64731">ξ</m:ci>
																	</m:apply>
																	<m:ci id="id64734">g</m:ci>
																</m:apply>
																<m:apply id="id64736">
																	<m:ci id="id64737">∈</m:ci>
																	<m:ci id="id64739">α</m:ci>
																	<m:apply id="id64742">
																		<m:csymbol id="id64743" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id64747">g</m:ci>
																		<m:times id="id64750"/>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id64752" class="eqpad"/>
									</tr>

								</table>
								<p id="id64758" style="" class="p">and demanding that <m:math id="id64765" display="inline">
										<m:semantics id="id64768">
											<m:mi id="id64769">η</m:mi>
											<m:annotation-xml id="id64771" encoding="MathML-Content">
												<m:ci id="id64775">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> be skew-symmetric. Applying this definition
fibrewise defines an invariant symplectic form on the vector bundle
									<m:math id="id64779" display="inline">
										<m:semantics id="id64783">
											<m:mrow id="id64784">
												<m:mrow id="id64785">
													<m:mi id="id64786">Z</m:mi>
													<m:mo id="id64788">×</m:mo>
													<m:mfenced id="id64790" open="(" close=")">
														<m:mrow id="id64795">
															<m:mi id="id64796" mathvariant="fraktur">g</m:mi>
															<m:mo id="id64801">⊕</m:mo>
															<m:msup id="id64803">
																<m:mi id="id64804" mathvariant="fraktur">g</m:mi>
																<m:mo id="id64809">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id64811">→</m:mo>
												<m:mi id="id64813">Z</m:mi>
											</m:mrow>
											<m:annotation-xml id="id64815" encoding="MathML-Content">
												<m:apply id="id64819">
													<m:ci id="id64820">→</m:ci>
													<m:apply id="id64822">
														<m:times id="id64823"/>
														<m:ci id="id64824">Z</m:ci>
														<m:apply id="id64826">
															<m:ci id="id64828">⊕</m:ci>
															<m:ci id="id64830">g</m:ci>
															<m:apply id="id64832">
																<m:csymbol id="id64833" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id64838">g</m:ci>
																<m:times id="id64840"/>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:ci id="id64841">Z</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which we will also denote by <m:math id="id64845" display="inline">
										<m:semantics id="id64848">
											<m:mi id="id64849">η</m:mi>
											<m:annotation-xml id="id64851" encoding="MathML-Content">
												<m:ci id="id64855">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
Then there exists a <m:math id="id64858" display="inline">
										<m:semantics id="id64862">
											<m:mi id="id64863">G</m:mi>
											<m:annotation-xml id="id64865" encoding="MathML-Content">
												<m:ci id="id64868">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant isomorphism of symplectic vector bundles
over <m:math id="id64872" display="inline">
										<m:semantics id="id64876">
											<m:mrow id="id64877">
												<m:mi id="id64878">Z</m:mi>
												<m:mo id="id64880">=</m:mo>
												<m:mrow id="id64882">
													<m:msubsup id="id64883">
														<m:mi id="id64884">μ</m:mi>
														<m:mi id="id64886">G</m:mi>
														<m:mrow id="id64889">
															<m:mo id="id64890">-</m:mo>
															<m:mn id="id64892">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id64894">⁢</m:mo>
													<m:mfenced id="id64896" open="(" close=")">
														<m:mn id="id64901">0</m:mn>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id64904" encoding="MathML-Content">
												<m:apply id="id64907">
													<m:eq id="id64908"/>
													<m:ci id="id64909">Z</m:ci>
													<m:apply id="id64911">
														<m:times id="id64912"/>
														<m:apply id="id64913">
															<m:csymbol id="id64914" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id64919">
																<m:csymbol id="id64920" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id64925">μ</m:ci>
																<m:ci id="id64927">G</m:ci>
															</m:apply>
															<m:apply id="id64929">
																<m:minus id="id64930"/>
																<m:cn id="id64931">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id64934">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
								</p>
								<table class="equation" id="S1.E3">

									<tr valign="baseline" class="equation" id="S1.E3">
										<td id="id64952" class="eqpad"/>
										<td id="id64957" nowrap="yes" align="center" colspan="1">
											<m:math id="id64964" display="block">
												<m:semantics id="id64967">
													<m:mrow id="id64968">
														<m:mi id="id64970">T</m:mi>
														<m:msub id="id64972">
															<m:mi id="id64973">X</m:mi>
															<m:mi id="id64975">Z</m:mi>
														</m:msub>
														<m:mo id="id64977">≅</m:mo>
														<m:mo id="id64979">(</m:mo>
														<m:mi id="id64982">Z</m:mi>
														<m:mo id="id64984">×</m:mo>
														<m:mi id="id64986" mathvariant="fraktur">g</m:mi>
														<m:mo id="id64990">)</m:mo>
														<m:mo id="id64993">⊕</m:mo>
														<m:mo id="id64995">(</m:mo>
														<m:mi id="id64997">Z</m:mi>
														<m:mo id="id64999">×</m:mo>
														<m:msup id="id65002">
															<m:mi id="id65003" mathvariant="fraktur">g</m:mi>
															<m:mo id="id65007">*</m:mo>
														</m:msup>
														<m:mo id="id65009">)</m:mo>
														<m:mo id="id65011">⊕</m:mo>
														<m:msubsup id="id65014">
															<m:mi id="id65015">π</m:mi>
															<m:mi id="id65017">G</m:mi>
															<m:mo id="id65019">*</m:mo>
														</m:msubsup>
														<m:mi id="id65022">T</m:mi>
														<m:mo id="id65024">(</m:mo>
														<m:mi id="id65026">X</m:mi>
														<m:mo id="id65028">/</m:mo>
														<m:mo id="id65030">/</m:mo>
														<m:mi id="id65032">G</m:mi>
														<m:mo id="id65034">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id65036" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id65041" class="eqpad"/>
										<td id="id65044" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id65055" class="refnum">1.3</span>)</td>
									</tr>

								</table>
								<p id="id65064" style="" class="p">where the symplectic form on the left is the restriction of
the symplectic form on <m:math id="id65070" display="inline">
										<m:semantics id="id65074">
											<m:mi id="id65075">X</m:mi>
											<m:annotation-xml id="id65077" encoding="MathML-Content">
												<m:ci id="id65080">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the symplectic form on the right
is the direct sum of the symplectic form on <m:math id="id65084" display="inline">
										<m:semantics id="id65088">
											<m:mrow id="id65089">
												<m:mfenced id="id65090" open="(" close=")">
													<m:mrow id="id65095">
														<m:mi id="id65096">Z</m:mi>
														<m:mo id="id65098">×</m:mo>
														<m:mi id="id65100" mathvariant="fraktur">g</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:mo id="id65105">⊕</m:mo>
												<m:mfenced id="id65107" open="(" close=")">
													<m:mrow id="id65112">
														<m:mi id="id65113">Z</m:mi>
														<m:mo id="id65116">×</m:mo>
														<m:msup id="id65118">
															<m:mi id="id65119" mathvariant="fraktur">g</m:mi>
															<m:mo id="id65123">*</m:mo>
														</m:msup>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id65126" encoding="MathML-Content">
												<m:apply id="id65129">
													<m:ci id="id65130">⊕</m:ci>
													<m:apply id="id65132">
														<m:times id="id65133"/>
														<m:ci id="id65134">Z</m:ci>
														<m:ci id="id65137">g</m:ci>
													</m:apply>
													<m:apply id="id65139">
														<m:times id="id65140"/>
														<m:ci id="id65141">Z</m:ci>
														<m:apply id="id65143">
															<m:csymbol id="id65144" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id65149">g</m:ci>
															<m:times id="id65151"/>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> given by <m:math id="id65153" display="inline">
										<m:semantics id="id65157">
											<m:mi id="id65158">η</m:mi>
											<m:annotation-xml id="id65160" encoding="MathML-Content">
												<m:ci id="id65163">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the pullback of the symplectic form on
									<m:math id="id65167" display="inline">
										<m:semantics id="id65171">
											<m:mrow id="id65172">
												<m:mi id="id65173">X</m:mi>
												<m:mo id="id65175">/</m:mo>
												<m:mo id="id65177">/</m:mo>
												<m:mi id="id65179">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id65181" encoding="MathML-Content">
												<m:ci id="id65185">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p13">
								<p id="id65197" style="" class="p">This isomorphism is defined as follows. A choice of connection on the
principal <m:math id="id65204" display="inline">
										<m:semantics id="id65207">
											<m:mi id="id65208">G</m:mi>
											<m:annotation-xml id="id65210" encoding="MathML-Content">
												<m:ci id="id65214">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-bundle <m:math id="id65217" display="inline">
										<m:semantics id="id65220">
											<m:mrow id="id65221">
												<m:mi id="id65222">Z</m:mi>
												<m:mo id="id65225">→</m:mo>
												<m:mi id="id65227">X</m:mi>
												<m:mo id="id65229">/</m:mo>
												<m:mo id="id65231">/</m:mo>
												<m:mi id="id65233">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id65236" encoding="MathML-Content">
												<m:ci id="id65239">Z</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> defines a ‘horizontal subbundle’
									<m:math id="id65243" display="inline">
										<m:semantics id="id65246">
											<m:mrow id="id65247">
												<m:mi id="id65248" mathvariant="script">H</m:mi>
												<m:mo id="id65253">⊂</m:mo>
												<m:mrow id="id65255">
													<m:mi id="id65256">T</m:mi>
													<m:mo id="id65258">⁢</m:mo>
													<m:mi id="id65261">Z</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id65263" encoding="MathML-Content">
												<m:apply id="id65266">
													<m:ci id="id65267">⊂</m:ci>
													<m:ci id="id65270">H</m:ci>
													<m:apply id="id65272">
														<m:times id="id65273"/>
														<m:ci id="id65274">T</m:ci>
														<m:ci id="id65276">Z</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which is isomorphic to <m:math id="id65280" display="inline">
										<m:semantics id="id65283">
											<m:mrow id="id65284">
												<m:msubsup id="id65285">
													<m:mi id="id65286">π</m:mi>
													<m:mi id="id65289">G</m:mi>
													<m:mo id="id65291">*</m:mo>
												</m:msubsup>
												<m:mi id="id65293">T</m:mi>
												<m:mo id="id65295">(</m:mo>
												<m:mi id="id65297">X</m:mi>
												<m:mo id="id65299">/</m:mo>
												<m:mo id="id65301">/</m:mo>
												<m:mi id="id65304">G</m:mi>
												<m:mo id="id65306">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id65308" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> (the
isomorphism is induced by the derivative <m:math id="id65313" display="inline">
										<m:semantics id="id65316">
											<m:mrow id="id65317">
												<m:mi id="id65318">d</m:mi>
												<m:mo id="id65320">⁢</m:mo>
												<m:msub id="id65323">
													<m:mi id="id65324">π</m:mi>
													<m:mi id="id65326">G</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id65328" encoding="MathML-Content">
												<m:apply id="id65332">
													<m:times id="id65333"/>
													<m:ci id="id65334">d</m:ci>
													<m:apply id="id65336">
														<m:csymbol id="id65337" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id65342">π</m:ci>
														<m:ci id="id65344">G</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>). Using the
inclusion <m:math id="id65348" display="inline">
										<m:semantics id="id65351">
											<m:mrow id="id65352">
												<m:mrow id="id65353">
													<m:mi id="id65354">T</m:mi>
													<m:mo id="id65356">⁢</m:mo>
													<m:mi id="id65359">Z</m:mi>
												</m:mrow>
												<m:mo id="id65361">↪</m:mo>
												<m:mrow id="id65363">
													<m:mi id="id65364">T</m:mi>
													<m:mo id="id65366">⁢</m:mo>
													<m:msub id="id65369">
														<m:mi id="id65370">X</m:mi>
														<m:mi id="id65372">Z</m:mi>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id65374" encoding="MathML-Content">
												<m:apply id="id65377">
													<m:ci id="id65378">↪</m:ci>
													<m:apply id="id65381">
														<m:times id="id65382"/>
														<m:ci id="id65383">T</m:ci>
														<m:ci id="id65385">Z</m:ci>
													</m:apply>
													<m:apply id="id65387">
														<m:times id="id65388"/>
														<m:ci id="id65389">T</m:ci>
														<m:apply id="id65392">
															<m:csymbol id="id65393" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id65397">X</m:ci>
															<m:ci id="id65399">Z</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> we can consider <m:math id="id65403" display="inline">
										<m:semantics id="id65406">
											<m:mi id="id65407" mathvariant="script">H</m:mi>
											<m:annotation-xml id="id65412" encoding="MathML-Content">
												<m:ci id="id65415">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to be a subbundle
of <m:math id="id65418" display="inline">
										<m:semantics id="id65422">
											<m:mrow id="id65423">
												<m:mi id="id65424">T</m:mi>
												<m:mo id="id65426">⁢</m:mo>
												<m:msub id="id65428">
													<m:mi id="id65430">X</m:mi>
													<m:mi id="id65432">Z</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id65434" encoding="MathML-Content">
												<m:apply id="id65437">
													<m:times id="id65438"/>
													<m:ci id="id65439">T</m:ci>
													<m:apply id="id65441">
														<m:csymbol id="id65442" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id65447">X</m:ci>
														<m:ci id="id65449">Z</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and we define <m:math id="id65453" display="inline">
										<m:semantics id="id65456">
											<m:mrow id="id65457">
												<m:msup id="id65458">
													<m:mi id="id65459" mathvariant="script">H</m:mi>
													<m:mi id="id65464">ω</m:mi>
												</m:msup>
												<m:mo id="id65466">⊂</m:mo>
												<m:mrow id="id65468">
													<m:mi id="id65470">T</m:mi>
													<m:mo id="id65472">⁢</m:mo>
													<m:msub id="id65474">
														<m:mi id="id65475">X</m:mi>
														<m:mi id="id65477">Z</m:mi>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id65479" encoding="MathML-Content">
												<m:apply id="id65483">
													<m:ci id="id65484">⊂</m:ci>
													<m:apply id="id65486">
														<m:csymbol id="id65487" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id65492">H</m:ci>
														<m:ci id="id65494">ω</m:ci>
													</m:apply>
													<m:apply id="id65496">
														<m:times id="id65498"/>
														<m:ci id="id65499">T</m:ci>
														<m:apply id="id65501">
															<m:csymbol id="id65502" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id65506">X</m:ci>
															<m:ci id="id65509">Z</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to be the
symplectic complement to <m:math id="id65512" display="inline">
										<m:semantics id="id65516">
											<m:mi id="id65517" mathvariant="script">H</m:mi>
											<m:annotation-xml id="id65521" encoding="MathML-Content">
												<m:ci id="id65524">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, with respect to the restriction of the
symplectic form <m:math id="id65528" display="inline">
										<m:semantics id="id65532">
											<m:mi id="id65533">ω</m:mi>
											<m:annotation-xml id="id65535" encoding="MathML-Content">
												<m:ci id="id65538">ω</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on <m:math id="id65542" display="inline">
										<m:semantics id="id65545">
											<m:mi id="id65546">X</m:mi>
											<m:annotation-xml id="id65549" encoding="MathML-Content">
												<m:ci id="id65552">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Standard calculations using the moment
map then imply (1) the restriction of <m:math id="id65556" display="inline">
										<m:semantics id="id65559">
											<m:mi id="id65560">ω</m:mi>
											<m:annotation-xml id="id65563" encoding="MathML-Content">
												<m:ci id="id65566">ω</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to <m:math id="id65570" display="inline">
										<m:semantics id="id65573">
											<m:mi id="id65574" mathvariant="script">H</m:mi>
											<m:annotation-xml id="id65579" encoding="MathML-Content">
												<m:ci id="id65582">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> equals the
pullback of the symplectic form on <m:math id="id65586" display="inline">
										<m:semantics id="id65589">
											<m:mrow id="id65590">
												<m:mi id="id65591">X</m:mi>
												<m:mo id="id65593">/</m:mo>
												<m:mo id="id65595">/</m:mo>
												<m:mi id="id65598">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id65600" encoding="MathML-Content">
												<m:ci id="id65603">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, (2) the subbundle
									<m:math id="id65606" display="inline">
										<m:semantics id="id65610">
											<m:msup id="id65611">
												<m:mi id="id65612" mathvariant="script">H</m:mi>
												<m:mi id="id65616">ω</m:mi>
											</m:msup>
											<m:annotation-xml id="id65619" encoding="MathML-Content">
												<m:apply id="id65622">
													<m:csymbol id="id65623" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id65628">H</m:ci>
													<m:ci id="id65630">ω</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a vector bundle complement to <m:math id="id65634" display="inline">
										<m:semantics id="id65637">
											<m:mi id="id65638" mathvariant="script">H</m:mi>
											<m:annotation-xml id="id65643" encoding="MathML-Content">
												<m:ci id="id65646">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, containing
									<m:math id="id65649" display="inline">
										<m:semantics id="id65653">
											<m:mrow id="id65654">
												<m:mrow id="id65655">
													<m:mrow id="id65656">
														<m:mi id="id65657">v</m:mi>
														<m:mo id="id65659">⁢</m:mo>
														<m:mi id="id65662">e</m:mi>
														<m:mo id="id65664">⁢</m:mo>
														<m:mi id="id65666">r</m:mi>
														<m:mo id="id65668">⁢</m:mo>
														<m:mi id="id65671">t</m:mi>
													</m:mrow>
													<m:mo id="id65673">⁢</m:mo>
													<m:mfenced id="id65675" open="(" close=")">
														<m:msub id="id65680">
															<m:mi id="id65681">π</m:mi>
															<m:mi id="id65684">G</m:mi>
														</m:msub>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id65686">≅</m:mo>
												<m:mfenced id="id65688" open="(" close=")">
													<m:mrow id="id65693">
														<m:mi id="id65694">Z</m:mi>
														<m:mo id="id65696">×</m:mo>
														<m:mi id="id65699" mathvariant="fraktur">g</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id65703" encoding="MathML-Content">
												<m:apply id="id65707">
													<m:approx id="id65708"/>
													<m:apply id="id65709">
														<m:times id="id65710"/>
														<m:ci id="id65711"/>
														<m:apply id="id65712">
															<m:csymbol id="id65713" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id65718">π</m:ci>
															<m:ci id="id65720">G</m:ci>
														</m:apply>
													</m:apply>
													<m:apply id="id65722">
														<m:times id="id65723"/>
														<m:ci id="id65724">Z</m:ci>
														<m:ci id="id65726">g</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and isomorphic to
									<m:math id="id65730" display="inline">
										<m:semantics id="id65733">
											<m:mrow id="id65734">
												<m:mfenced id="id65735" open="(" close=")">
													<m:mrow id="id65740">
														<m:mi id="id65742">Z</m:mi>
														<m:mo id="id65744">×</m:mo>
														<m:mi id="id65746" mathvariant="fraktur">g</m:mi>
													</m:mrow>
												</m:mfenced>
												<m:mo id="id65750">⊕</m:mo>
												<m:mfenced id="id65753" open="(" close=")">
													<m:mrow id="id65758">
														<m:mi id="id65759">Z</m:mi>
														<m:mo id="id65761">×</m:mo>
														<m:msup id="id65764">
															<m:mi id="id65765" mathvariant="fraktur">g</m:mi>
															<m:mo id="id65769">*</m:mo>
														</m:msup>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id65771" encoding="MathML-Content">
												<m:apply id="id65774">
													<m:ci id="id65776">⊕</m:ci>
													<m:apply id="id65778">
														<m:times id="id65779"/>
														<m:ci id="id65780">Z</m:ci>
														<m:ci id="id65782">g</m:ci>
													</m:apply>
													<m:apply id="id65784">
														<m:times id="id65785"/>
														<m:ci id="id65786">Z</m:ci>
														<m:apply id="id65789">
															<m:csymbol id="id65790" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id65794">g</m:ci>
															<m:times id="id65796"/>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (with the isomorphism given by choosing an
equivariant complement to <m:math id="id65799" display="inline">
										<m:semantics id="id65803">
											<m:mrow id="id65804">
												<m:mrow id="id65805">
													<m:mi id="id65806">v</m:mi>
													<m:mo id="id65808">⁢</m:mo>
													<m:mi id="id65810">e</m:mi>
													<m:mo id="id65812">⁢</m:mo>
													<m:mi id="id65815">r</m:mi>
													<m:mo id="id65817">⁢</m:mo>
													<m:mi id="id65819">t</m:mi>
												</m:mrow>
												<m:mo id="id65822">⁢</m:mo>
												<m:mfenced id="id65824" open="(" close=")">
													<m:msub id="id65829">
														<m:mi id="id65830">π</m:mi>
														<m:mi id="id65832">G</m:mi>
													</m:msub>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id65835" encoding="MathML-Content">
												<m:apply id="id65838">
													<m:times id="id65839"/>
													<m:ci id="id65840"/>
													<m:apply id="id65841">
														<m:csymbol id="id65842" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id65847">π</m:ci>
														<m:ci id="id65849">G</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and identifying this
complement with <m:math id="id65853" display="inline">
										<m:semantics id="id65856">
											<m:mrow id="id65857">
												<m:mi id="id65858">Z</m:mi>
												<m:mo id="id65860">×</m:mo>
												<m:msup id="id65863">
													<m:mi id="id65864" mathvariant="fraktur">g</m:mi>
													<m:mo id="id65868">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id65870" encoding="MathML-Content">
												<m:apply id="id65874">
													<m:times id="id65875"/>
													<m:ci id="id65876">Z</m:ci>
													<m:apply id="id65878">
														<m:csymbol id="id65879" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id65884">g</m:ci>
														<m:times id="id65886"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> via <m:math id="id65888" display="inline">
										<m:semantics id="id65892">
											<m:mrow id="id65893">
												<m:mi id="id65894">d</m:mi>
												<m:mo id="id65896">⁢</m:mo>
												<m:mi id="id65898">μ</m:mi>
											</m:mrow>
											<m:annotation-xml id="id65901" encoding="MathML-Content">
												<m:apply id="id65904">
													<m:times id="id65905"/>
													<m:ci id="id65906">d</m:ci>
													<m:ci id="id65908">μ</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>), and (3) the restriction of
									<m:math id="id65912" display="inline">
										<m:semantics id="id65916">
											<m:mi id="id65917">ω</m:mi>
											<m:annotation-xml id="id65919" encoding="MathML-Content">
												<m:ci id="id65922">ω</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to <m:math id="id65926" display="inline">
										<m:semantics id="id65929">
											<m:mrow id="id65930">
												<m:msup id="id65932">
													<m:mi id="id65933" mathvariant="script">H</m:mi>
													<m:mi id="id65937">ω</m:mi>
												</m:msup>
												<m:mo id="id65939">≅</m:mo>
												<m:mrow id="id65942">
													<m:mfenced id="id65943" open="(" close=")">
														<m:mrow id="id65948">
															<m:mi id="id65949">Z</m:mi>
															<m:mo id="id65951">×</m:mo>
															<m:mi id="id65954" mathvariant="fraktur">g</m:mi>
														</m:mrow>
													</m:mfenced>
													<m:mo id="id65958">⊕</m:mo>
													<m:mfenced id="id65960" open="(" close=")">
														<m:mrow id="id65965">
															<m:mi id="id65966">Z</m:mi>
															<m:mo id="id65969">×</m:mo>
															<m:msup id="id65971">
																<m:mi id="id65972" mathvariant="fraktur">g</m:mi>
																<m:mo id="id65976">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id65979" encoding="MathML-Content">
												<m:apply id="id65982">
													<m:approx id="id65983"/>
													<m:apply id="id65984">
														<m:csymbol id="id65985" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id65990">H</m:ci>
														<m:ci id="id65992">ω</m:ci>
													</m:apply>
													<m:apply id="id65994">
														<m:ci id="id65995">⊕</m:ci>
														<m:apply id="id65998">
															<m:times id="id65999"/>
															<m:ci id="id66000">Z</m:ci>
															<m:ci id="id66002">g</m:ci>
														</m:apply>
														<m:apply id="id66004">
															<m:times id="id66005"/>
															<m:ci id="id66006">Z</m:ci>
															<m:apply id="id66008">
																<m:csymbol id="id66010" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id66014">g</m:ci>
																<m:times id="id66016"/>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> equals the symplectic
form <m:math id="id66019" display="inline">
										<m:semantics id="id66022">
											<m:mi id="id66023">η</m:mi>
											<m:annotation-xml id="id66026" encoding="MathML-Content">
												<m:ci id="id66029">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> defined above.</p>
							</div>

							<div class="para" id="S1.p14">
								<p id="id66042" style="" class="p">The same arguments, applied to  <m:math id="id66048" display="inline">
										<m:semantics id="id66051">
											<m:mi id="id66052">T</m:mi>
											<m:annotation-xml id="id66055" encoding="MathML-Content">
												<m:ci id="id66058">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id66061" display="inline">
										<m:semantics id="id66065">
											<m:mrow id="id66066">
												<m:msubsup id="id66067">
													<m:mi id="id66068">μ</m:mi>
													<m:mi id="id66070">T</m:mi>
													<m:mrow id="id66072">
														<m:mo id="id66074">-</m:mo>
														<m:mn id="id66076">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id66078">⁢</m:mo>
												<m:mfenced id="id66080" open="(" close=")">
													<m:mn id="id66085">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id66087" encoding="MathML-Content">
												<m:apply id="id66091">
													<m:times id="id66092"/>
													<m:apply id="id66093">
														<m:csymbol id="id66094" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id66099">
															<m:csymbol id="id66100" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id66104">μ</m:ci>
															<m:ci id="id66107">T</m:ci>
														</m:apply>
														<m:apply id="id66109">
															<m:minus id="id66110"/>
															<m:cn id="id66111">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id66113">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in place
of <m:math id="id66117" display="inline">
										<m:semantics id="id66120">
											<m:mi id="id66121">G</m:mi>
											<m:annotation-xml id="id66123" encoding="MathML-Content">
												<m:ci id="id66126">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id66130" display="inline">
										<m:semantics id="id66133">
											<m:mrow id="id66134">
												<m:mi id="id66135">Z</m:mi>
												<m:mo id="id66138">=</m:mo>
												<m:mrow id="id66140">
													<m:msubsup id="id66141">
														<m:mi id="id66142">μ</m:mi>
														<m:mi id="id66144">G</m:mi>
														<m:mrow id="id66146">
															<m:mo id="id66147">-</m:mo>
															<m:mn id="id66150">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id66152">⁢</m:mo>
													<m:mfenced id="id66154" open="(" close=")">
														<m:mn id="id66159">0</m:mn>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id66161" encoding="MathML-Content">
												<m:apply id="id66165">
													<m:eq id="id66166"/>
													<m:ci id="id66167">Z</m:ci>
													<m:apply id="id66169">
														<m:times id="id66170"/>
														<m:apply id="id66171">
															<m:csymbol id="id66172" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id66177">
																<m:csymbol id="id66178" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id66182">μ</m:ci>
																<m:ci id="id66185">G</m:ci>
															</m:apply>
															<m:apply id="id66187">
																<m:minus id="id66188"/>
																<m:cn id="id66189">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id66191">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, give an analogous isomorphism to
that of equation (<a id="id66197" href="#S1.E3" title="1.3 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id66196" style="" class="text tag">1.3</span>
									</a>) above. Combining these two
isomorphisms, in a neighbourhood of <m:math id="id66213" display="inline">
										<m:semantics id="id66216">
											<m:mi id="id66217">Z</m:mi>
											<m:annotation-xml id="id66219" encoding="MathML-Content">
												<m:ci id="id66222">Z</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and arguing as in step 2 of
this proof gives an isomorphism of symplectic vector bundles</p>
								<table class="equation" id="S1.E4">

									<tr valign="baseline" class="equation" id="S1.E4">
										<td id="id66243" class="eqpad"/>
										<td id="id66247" nowrap="yes" align="center" colspan="1">
											<m:math id="id66254" display="block">
												<m:semantics id="id66258">
													<m:mrow id="id66259">
														<m:mi id="id66260">T</m:mi>
														<m:mo id="id66262">(</m:mo>
														<m:mi id="id66264">X</m:mi>
														<m:mo id="id66266">/</m:mo>
														<m:mo id="id66268">/</m:mo>
														<m:mi id="id66271">T</m:mi>
														<m:msub id="id66273">
															<m:mo id="id66274">)</m:mo>
															<m:mrow id="id66276">
																<m:mi id="id66277">Z</m:mi>
																<m:mo id="id66279">/</m:mo>
																<m:mi id="id66281">T</m:mi>
															</m:mrow>
														</m:msub>
														<m:mo id="id66283">≅</m:mo>
														<m:mo id="id66286">(</m:mo>
														<m:mi id="id66288">Z</m:mi>
														<m:msub id="id66290">
															<m:mo id="id66291">×</m:mo>
															<m:mi id="id66294">T</m:mi>
														</m:msub>
														<m:mi id="id66296" mathvariant="fraktur">v</m:mi>
														<m:mo id="id66300">)</m:mo>
														<m:mo id="id66302">⊕</m:mo>
														<m:mo id="id66305">(</m:mo>
														<m:mi id="id66307">Z</m:mi>
														<m:msub id="id66309">
															<m:mo id="id66310">×</m:mo>
															<m:mi id="id66312">T</m:mi>
														</m:msub>
														<m:msup id="id66314">
															<m:mi id="id66316" mathvariant="fraktur">v</m:mi>
															<m:mo id="id66320">*</m:mo>
														</m:msup>
														<m:mo id="id66322">)</m:mo>
														<m:mo id="id66324">⊕</m:mo>
														<m:msup id="id66327">
															<m:mi id="id66328">π</m:mi>
															<m:mo id="id66330">*</m:mo>
														</m:msup>
														<m:mi id="id66332">T</m:mi>
														<m:mo id="id66334">(</m:mo>
														<m:mi id="id66336">X</m:mi>
														<m:mo id="id66339">/</m:mo>
														<m:mo id="id66341">/</m:mo>
														<m:mi id="id66343">G</m:mi>
														<m:mo id="id66345">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id66347" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id66352" class="eqpad"/>
										<td id="id66355" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id66366" class="refnum">1.4</span>)</td>
									</tr>

								</table>
								<p id="id66374" style="" class="p">such that the symplectic form on the left is the restriction of the
symplectic form on <m:math id="id66381" display="inline">
										<m:semantics id="id66384">
											<m:mrow id="id66386">
												<m:mi id="id66387">X</m:mi>
												<m:mo id="id66389">/</m:mo>
												<m:mo id="id66391">/</m:mo>
												<m:mi id="id66393">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id66395" encoding="MathML-Content">
												<m:ci id="id66398">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and on the right is the direct sum of the
natural symplectic form on <m:math id="id66402" display="inline">
										<m:semantics id="id66406">
											<m:mrow id="id66407">
												<m:mo id="id66408">(</m:mo>
												<m:mi id="id66410">Z</m:mi>
												<m:msub id="id66412">
													<m:mo id="id66413">×</m:mo>
													<m:mi id="id66416">T</m:mi>
												</m:msub>
												<m:mi id="id66418" mathvariant="fraktur">v</m:mi>
												<m:mo id="id66422">)</m:mo>
												<m:mo id="id66424">⊕</m:mo>
												<m:mo id="id66427">(</m:mo>
												<m:mi id="id66429">Z</m:mi>
												<m:msub id="id66431">
													<m:mo id="id66432">×</m:mo>
													<m:mi id="id66434">T</m:mi>
												</m:msub>
												<m:msup id="id66436">
													<m:mi id="id66438" mathvariant="fraktur">v</m:mi>
													<m:mo id="id66442">*</m:mo>
												</m:msup>
												<m:mo id="id66444">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id66446" encoding="MathML-Content">
												<m:ci id="id66450">(</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> defined
analogously to <m:math id="id66453" display="inline">
										<m:semantics id="id66456">
											<m:mi id="id66458">η</m:mi>
											<m:annotation-xml id="id66460" encoding="MathML-Content">
												<m:ci id="id66463">η</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the pullback of the symplectic form on
									<m:math id="id66467" display="inline">
										<m:semantics id="id66471">
											<m:mrow id="id66472">
												<m:mi id="id66473">X</m:mi>
												<m:mo id="id66475">/</m:mo>
												<m:mo id="id66477">/</m:mo>
												<m:mi id="id66479">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id66481" encoding="MathML-Content">
												<m:ci id="id66485">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p15">
								<p id="id66497" style="" class="p">
									<span id="id66502" style="" class="text italic">3(ii) The almost complex structure <m:math id="id66508" display="inline">
											<m:semantics id="id66512">
												<m:msub id="id66513">
													<m:mover id="id66514" accent="true">
														<m:mi id="id66517">J</m:mi>
														<m:mo id="id66519" mathvariant="normal">~</m:mo>
													</m:mover>
													<m:mn id="id66524" mathvariant="normal">0</m:mn>
												</m:msub>
												<m:annotation-xml id="id66528" encoding="MathML-Content">
													<m:apply id="id66532">
														<m:csymbol id="id66533" cd="ambiguous">subscript</m:csymbol>
														<m:apply id="id66537">
															<m:ci id="id66538">~</m:ci>
															<m:ci id="id66540">J</m:ci>
														</m:apply>
														<m:cn id="id66543">0</m:cn>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</span>
								</p>
							</div>

							<div class="para" id="S1.p16">
								<p id="id66555" style="" class="p">Fix (1) an almost complex structure on <m:math id="id66561" display="inline">
										<m:semantics id="id66564">
											<m:mrow id="id66566">
												<m:mi id="id66567">X</m:mi>
												<m:mo id="id66569">/</m:mo>
												<m:mo id="id66571">/</m:mo>
												<m:mi id="id66573">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id66575" encoding="MathML-Content">
												<m:ci id="id66578">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> which is
compatible with the symplectic form,
and (2) a <m:math id="id66582" display="inline">
										<m:semantics id="id66586">
											<m:mi id="id66587">T</m:mi>
											<m:annotation-xml id="id66589" encoding="MathML-Content">
												<m:ci id="id66592">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-invariant positive-definite
inner product on <m:math id="id66596" display="inline">
										<m:semantics id="id66599">
											<m:mi id="id66600" mathvariant="fraktur">v</m:mi>
											<m:annotation-xml id="id66605" encoding="MathML-Content">
												<m:ci id="id66608">v</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p17">
								<p id="id66620" style="" class="p">The inner product on <m:math id="id66626" display="inline">
										<m:semantics id="id66630">
											<m:mi id="id66631" mathvariant="fraktur">v</m:mi>
											<m:annotation-xml id="id66635" encoding="MathML-Content">
												<m:ci id="id66639">v</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> gives a duality
isomorphism <m:math id="id66642" display="inline">
										<m:semantics id="id66646">
											<m:mrow id="id66647">
												<m:mi id="id66648" mathvariant="fraktur">v</m:mi>
												<m:mstyle id="id66686" scriptlevel="+1">
													<m:mtable id="id66690" rowspacing="0.2ex" columnspacing="0.4em">
														<m:mtr id="id66695">
															<m:mtd id="id66697">
																<m:mo id="id66698">≅</m:mo>
															</m:mtd>
														</m:mtr>
														<m:mtr id="id66700">
															<m:mtd id="id66701">
																<m:mi id="id66702">→</m:mi>
															</m:mtd>
														</m:mtr>
													</m:mtable>
												</m:mstyle>
												<m:msup id="id66705">
													<m:mi id="id66706" mathvariant="fraktur">v</m:mi>
													<m:mo id="id66710">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id66712" encoding="MathML-Content">
												<m:apply id="id66716">
													<m:apply id="id66717">
														<m:ci id="id66718"/>
														<m:approx id="id66719"/>
														<m:ci id="id66720">→</m:ci>
													</m:apply>
													<m:ci id="id66722">v</m:ci>
													<m:apply id="id66724">
														<m:csymbol id="id66726" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id66730">v</m:ci>
														<m:times id="id66732"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which is <m:math id="id66735" display="inline">
										<m:semantics id="id66738">
											<m:mi id="id66739">T</m:mi>
											<m:annotation-xml id="id66741" encoding="MathML-Content">
												<m:ci id="id66745">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant, and which thus
descends to an isomorphism</p>
								<table class="equation" id="S1.Ex11">

									<tr valign="baseline" class="equation" id="S1.Ex11">
										<td id="id66765" class="eqpad"/>
										<td id="id66769" nowrap="yes" align="center" colspan="1">
											<m:math id="id66777" display="block">
												<m:semantics id="id66780">
													<m:mrow id="id66781">
														<m:mi id="id66782">φ</m:mi>
														<m:mo id="id66785">:</m:mo>
														<m:mi id="id66787">Z</m:mi>
														<m:msub id="id66789">
															<m:mo id="id66790">×</m:mo>
															<m:mi id="id66792">T</m:mi>
														</m:msub>
														<m:mi id="id66794" mathvariant="fraktur">v</m:mi>
														<m:mo id="id66799">→</m:mo>
														<m:mi id="id66801">Z</m:mi>
														<m:msub id="id66803">
															<m:mo id="id66804">×</m:mo>
															<m:mi id="id66807">T</m:mi>
														</m:msub>
														<m:msup id="id66809">
															<m:mi id="id66810" mathvariant="fraktur">v</m:mi>
															<m:mo id="id66814">*</m:mo>
														</m:msup>
														<m:mo id="id66817">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id66819" encoding="MathML-Content">
														<m:ci id="id66822">φ</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id66826" class="eqpad"/>
									</tr>

								</table>
								<p id="id66832" style="" class="p">We now define <m:math id="id66838" display="inline">
										<m:semantics id="id66842">
											<m:msub id="id66843">
												<m:mi id="id66844">J</m:mi>
												<m:mn id="id66846">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id66848" encoding="MathML-Content">
												<m:apply id="id66852">
													<m:csymbol id="id66853" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id66857">J</m:ci>
													<m:cn id="id66859">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. On the subbundle <m:math id="id66863" display="inline">
										<m:semantics id="id66866">
											<m:mrow id="id66867">
												<m:msup id="id66868">
													<m:mi id="id66869">π</m:mi>
													<m:mo id="id66872">*</m:mo>
												</m:msup>
												<m:mi id="id66874">T</m:mi>
												<m:mo id="id66876">(</m:mo>
												<m:mi id="id66878">X</m:mi>
												<m:mo id="id66880">/</m:mo>
												<m:mo id="id66882">/</m:mo>
												<m:mi id="id66885">G</m:mi>
												<m:mo id="id66887">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id66889" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> we define <m:math id="id66894" display="inline">
										<m:semantics id="id66897">
											<m:msub id="id66898">
												<m:mi id="id66899">J</m:mi>
												<m:mn id="id66901">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id66903" encoding="MathML-Content">
												<m:apply id="id66907">
													<m:csymbol id="id66908" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id66912">J</m:ci>
													<m:cn id="id66914">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
to equal the almost complex structure on <m:math id="id66918" display="inline">
										<m:semantics id="id66922">
											<m:mrow id="id66923">
												<m:mi id="id66924">X</m:mi>
												<m:mo id="id66926">/</m:mo>
												<m:mo id="id66928">/</m:mo>
												<m:mi id="id66930">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id66932" encoding="MathML-Content">
												<m:ci id="id66936">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> which we fixed
above. On the subbundle <m:math id="id66939" display="inline">
										<m:semantics id="id66943">
											<m:mrow id="id66944">
												<m:mo id="id66945">(</m:mo>
												<m:mi id="id66947">Z</m:mi>
												<m:msub id="id66949">
													<m:mo id="id66950">×</m:mo>
													<m:mi id="id66952">T</m:mi>
												</m:msub>
												<m:mi id="id66955" mathvariant="fraktur">v</m:mi>
												<m:mo id="id66959">)</m:mo>
												<m:mo id="id66961">⊕</m:mo>
												<m:mo id="id66964">(</m:mo>
												<m:mi id="id66966">Z</m:mi>
												<m:msub id="id66968">
													<m:mo id="id66969">×</m:mo>
													<m:mi id="id66971">T</m:mi>
												</m:msub>
												<m:msup id="id66973">
													<m:mi id="id66974" mathvariant="fraktur">v</m:mi>
													<m:mo id="id66979">*</m:mo>
												</m:msup>
												<m:mo id="id66981">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id66983" encoding="MathML-Content">
												<m:ci id="id66986">(</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> we define <m:math id="id66990" display="inline">
										<m:semantics id="id66993">
											<m:msub id="id66994">
												<m:mi id="id66995">J</m:mi>
												<m:mn id="id66998">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id67000" encoding="MathML-Content">
												<m:apply id="id67003">
													<m:csymbol id="id67004" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67009">J</m:ci>
													<m:cn id="id67011">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
to equal <m:math id="id67014" display="inline">
										<m:semantics id="id67018">
											<m:mfenced id="id67019" open="(" close=")">
												<m:mtable id="id67024" rowspacing="0.2ex" columnspacing="0.4em">
													<m:mtr id="id67029">
														<m:mtd id="id67030" columnalign="center">
															<m:mn id="id67034">0</m:mn>
														</m:mtd>
														<m:mtd id="id67036" columnalign="center">
															<m:mrow id="id67039">
																<m:mo id="id67040">-</m:mo>
																<m:mi id="id67042">φ</m:mi>
															</m:mrow>
														</m:mtd>
													</m:mtr>
													<m:mtr id="id67045">
														<m:mtd id="id67046" columnalign="center">
															<m:mi id="id67049">φ</m:mi>
														</m:mtd>
														<m:mtd id="id67052" columnalign="center">
															<m:mn id="id67055">0</m:mn>
														</m:mtd>
													</m:mtr>
												</m:mtable>
											</m:mfenced>
											<m:annotation-xml id="id67057" encoding="MathML-Content">
												<m:mtext id="id67060">0-φφ0</m:mtext>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.  One easily
checks that <m:math id="id67064" display="inline">
										<m:semantics id="id67068">
											<m:msub id="id67069">
												<m:mi id="id67070">J</m:mi>
												<m:mn id="id67072">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id67074" encoding="MathML-Content">
												<m:apply id="id67077">
													<m:csymbol id="id67078" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67083">J</m:ci>
													<m:cn id="id67085">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is compatible with the symplectic form on
									<m:math id="id67089" display="inline">
										<m:semantics id="id67092">
											<m:mrow id="id67093">
												<m:mi id="id67094">T</m:mi>
												<m:mo id="id67097">(</m:mo>
												<m:mi id="id67099">X</m:mi>
												<m:mo id="id67101">/</m:mo>
												<m:mo id="id67103">/</m:mo>
												<m:mi id="id67105">T</m:mi>
												<m:msub id="id67107">
													<m:mo id="id67108">)</m:mo>
													<m:mrow id="id67110">
														<m:mi id="id67112">Z</m:mi>
														<m:mo id="id67114">/</m:mo>
														<m:mi id="id67116">T</m:mi>
													</m:mrow>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id67118" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>, and it follows from standard results in
symplectic geometry that there exists an almost complex structure
									<m:math id="id67123" display="inline">
										<m:semantics id="id67127">
											<m:mover id="id67128" accent="true">
												<m:msub id="id67131">
													<m:mi id="id67132">J</m:mi>
													<m:mn id="id67134">0</m:mn>
												</m:msub>
												<m:mo id="id67136">~</m:mo>
											</m:mover>
											<m:annotation-xml id="id67139" encoding="MathML-Content">
												<m:apply id="id67142">
													<m:ci id="id67143">~</m:ci>
													<m:apply id="id67145">
														<m:csymbol id="id67146" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id67151">J</m:ci>
														<m:cn id="id67153">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on <m:math id="id67156" display="inline">
										<m:semantics id="id67160">
											<m:mrow id="id67161">
												<m:mi id="id67162">X</m:mi>
												<m:mo id="id67164">/</m:mo>
												<m:mo id="id67166">/</m:mo>
												<m:mi id="id67168">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id67170" encoding="MathML-Content">
												<m:ci id="id67174">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> which is compatible with the symplectic form,
and whose restriction equals <m:math id="id67178" display="inline">
										<m:semantics id="id67181">
											<m:msub id="id67182">
												<m:mi id="id67183">J</m:mi>
												<m:mn id="id67185">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id67188" encoding="MathML-Content">
												<m:apply id="id67191">
													<m:csymbol id="id67192" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67197">J</m:ci>
													<m:cn id="id67199">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (see for example McDuff and
Salamon <cite id="id67203" class="cite">[<a id="id67207" href="#bib.bib17" title="" class="ref">17</a>, proposition 4.1]</cite>).</p>
							</div>

							<div class="para" id="S1.p18">
								<p id="id67227" style="" class="p">
									<span id="id67232" style="" class="text italic">3(iii) The homotopy.</span>
								</p>
							</div>

							<div class="para" id="S1.p19">
								<p id="id67247" style="" class="p">Fix a choice of positive roots
									<m:math id="id67254" display="inline">
										<m:semantics id="id67257">
											<m:mrow id="id67258">
												<m:msup id="id67259">
													<m:mi id="id67260" mathvariant="normal">Δ</m:mi>
													<m:mo id="id67265">+</m:mo>
												</m:msup>
												<m:mo id="id67267">⊂</m:mo>
												<m:mi id="id67270" mathvariant="normal">Δ</m:mi>
											</m:mrow>
											<m:annotation-xml id="id67274" encoding="MathML-Content">
												<m:apply id="id67278">
													<m:ci id="id67279">⊂</m:ci>
													<m:apply id="id67281">
														<m:csymbol id="id67282" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id67287">Δ</m:ci>
														<m:plus id="id67289"/>
													</m:apply>
													<m:ci id="id67290">Δ</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
This choice of positive roots gives a <m:math id="id67294" display="inline">
										<m:semantics id="id67298">
											<m:mi id="id67299">T</m:mi>
											<m:annotation-xml id="id67301" encoding="MathML-Content">
												<m:ci id="id67304">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-invariant complex structure
on <m:math id="id67308" display="inline">
										<m:semantics id="id67311">
											<m:mi id="id67312" mathvariant="fraktur">v</m:mi>
											<m:annotation-xml id="id67316" encoding="MathML-Content">
												<m:ci id="id67320">v</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which descends to a complex structure on <m:math id="id67324" display="inline">
										<m:semantics id="id67327">
											<m:mrow id="id67328">
												<m:mi id="id67329">Z</m:mi>
												<m:msub id="id67331">
													<m:mo id="id67332">×</m:mo>
													<m:mi id="id67335">T</m:mi>
												</m:msub>
												<m:mi id="id67337" mathvariant="fraktur">v</m:mi>
											</m:mrow>
											<m:annotation-xml id="id67341" encoding="MathML-Content">
												<m:ci id="id67344">Z</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which we
denote by <m:math id="id67348" display="inline">
										<m:semantics id="id67351">
											<m:msub id="id67352">
												<m:mi id="id67353">i</m:mi>
												<m:mi id="id67356" mathvariant="fraktur">v</m:mi>
											</m:msub>
											<m:annotation-xml id="id67360" encoding="MathML-Content">
												<m:apply id="id67363">
													<m:csymbol id="id67364" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67369">i</m:ci>
													<m:ci id="id67371">v</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Similarly, the negative roots define a complex
structure <m:math id="id67375" display="inline">
										<m:semantics id="id67378">
											<m:msub id="id67379">
												<m:mi id="id67380">i</m:mi>
												<m:msup id="id67383">
													<m:mi id="id67384" mathvariant="fraktur">v</m:mi>
													<m:mo id="id67388">*</m:mo>
												</m:msup>
											</m:msub>
											<m:annotation-xml id="id67390" encoding="MathML-Content">
												<m:apply id="id67394">
													<m:csymbol id="id67395" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67399">i</m:ci>
													<m:apply id="id67401">
														<m:csymbol id="id67402" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id67407">v</m:ci>
														<m:times id="id67409"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on <m:math id="id67412" display="inline">
										<m:semantics id="id67415">
											<m:mrow id="id67416">
												<m:mi id="id67417">Z</m:mi>
												<m:msub id="id67419">
													<m:mo id="id67420">×</m:mo>
													<m:mi id="id67423">T</m:mi>
												</m:msub>
												<m:msup id="id67425">
													<m:mi id="id67426" mathvariant="fraktur">v</m:mi>
													<m:mo id="id67430">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id67432" encoding="MathML-Content">
												<m:ci id="id67436">Z</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and we have</p>
								<table class="equation" id="S1.Ex12">

									<tr valign="baseline" class="equation" id="S1.Ex12">
										<td id="id67456" class="eqpad"/>
										<td id="id67460" nowrap="yes" align="center" colspan="1">
											<m:math id="id67468" display="block">
												<m:semantics id="id67471">
													<m:mrow id="id67472">
														<m:mrow id="id67473">
															<m:msub id="id67474">
																<m:mi id="id67475">i</m:mi>
																<m:msup id="id67477">
																	<m:mi id="id67478" mathvariant="fraktur">v</m:mi>
																	<m:mo id="id67483">*</m:mo>
																</m:msup>
															</m:msub>
															<m:mo id="id67485">=</m:mo>
															<m:mrow id="id67487">
																<m:mi id="id67488">φ</m:mi>
																<m:mo id="id67491">∘</m:mo>
																<m:mfenced id="id67493" open="(" close=")">
																	<m:mrow id="id67498">
																		<m:mo id="id67499">-</m:mo>
																		<m:msub id="id67501">
																			<m:mi id="id67502">i</m:mi>
																			<m:msup id="id67504">
																				<m:mi id="id67506" mathvariant="fraktur">v</m:mi>
																				<m:mo id="id67510">*</m:mo>
																			</m:msup>
																		</m:msub>
																	</m:mrow>
																</m:mfenced>
																<m:mo id="id67512">∘</m:mo>
																<m:msup id="id67514">
																	<m:mi id="id67516">φ</m:mi>
																	<m:mrow id="id67518">
																		<m:mo id="id67519">-</m:mo>
																		<m:mn id="id67521">1</m:mn>
																	</m:mrow>
																</m:msup>
															</m:mrow>
														</m:mrow>
														<m:mo id="id67523">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id67525" encoding="MathML-Content">
														<m:apply id="id67529">
															<m:eq id="id67530"/>
															<m:apply id="id67531">
																<m:csymbol id="id67532" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id67537">i</m:ci>
																<m:apply id="id67539">
																	<m:csymbol id="id67540" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id67544">v</m:ci>
																	<m:times id="id67547"/>
																</m:apply>
															</m:apply>
															<m:apply id="id67548">
																<m:compose id="id67549"/>
																<m:ci id="id67550">φ</m:ci>
																<m:apply id="id67552">
																	<m:minus id="id67553"/>
																	<m:apply id="id67554">
																		<m:csymbol id="id67555" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id67560">i</m:ci>
																		<m:apply id="id67562">
																			<m:csymbol id="id67563" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id67568">v</m:ci>
																			<m:times id="id67570"/>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id67571">
																	<m:csymbol id="id67572" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id67577">φ</m:ci>
																	<m:apply id="id67579">
																		<m:minus id="id67580"/>
																		<m:cn id="id67581">1</m:cn>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id67585" class="eqpad"/>
									</tr>

								</table>
								<p id="id67591" style="" class="p">We now define <m:math id="id67598" display="inline">
										<m:semantics id="id67601">
											<m:msub id="id67602">
												<m:mi id="id67603">J</m:mi>
												<m:mn id="id67605">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id67607" encoding="MathML-Content">
												<m:apply id="id67611">
													<m:csymbol id="id67612" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67616">J</m:ci>
													<m:cn id="id67618">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. On the subbundle <m:math id="id67622" display="inline">
										<m:semantics id="id67625">
											<m:mrow id="id67626">
												<m:msup id="id67627">
													<m:mi id="id67628">π</m:mi>
													<m:mo id="id67631">*</m:mo>
												</m:msup>
												<m:mi id="id67633">T</m:mi>
												<m:mo id="id67635">(</m:mo>
												<m:mi id="id67637">X</m:mi>
												<m:mo id="id67639">/</m:mo>
												<m:mo id="id67642">/</m:mo>
												<m:mi id="id67644">G</m:mi>
												<m:mo id="id67646">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id67648" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> we
define <m:math id="id67653" display="inline">
										<m:semantics id="id67656">
											<m:msub id="id67657">
												<m:mi id="id67658">J</m:mi>
												<m:mn id="id67660">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id67662" encoding="MathML-Content">
												<m:apply id="id67666">
													<m:csymbol id="id67667" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67671">J</m:ci>
													<m:cn id="id67674">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to equal the almost complex structure on
									<m:math id="id67677" display="inline">
										<m:semantics id="id67681">
											<m:mrow id="id67682">
												<m:mi id="id67683">X</m:mi>
												<m:mo id="id67685">/</m:mo>
												<m:mo id="id67687">/</m:mo>
												<m:mi id="id67689">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id67691" encoding="MathML-Content">
												<m:ci id="id67695">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> which we fixed above, and hence to agree with <m:math id="id67698" display="inline">
										<m:semantics id="id67702">
											<m:msub id="id67703">
												<m:mi id="id67704">J</m:mi>
												<m:mn id="id67706">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id67708" encoding="MathML-Content">
												<m:apply id="id67711">
													<m:csymbol id="id67712" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67717">J</m:ci>
													<m:cn id="id67719">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. On the subbundle
									<m:math id="id67723" display="inline">
										<m:semantics id="id67726">
											<m:mrow id="id67727">
												<m:mo id="id67728">(</m:mo>
												<m:mi id="id67730">Z</m:mi>
												<m:msub id="id67732">
													<m:mo id="id67734">×</m:mo>
													<m:mi id="id67736">T</m:mi>
												</m:msub>
												<m:mi id="id67738" mathvariant="fraktur">v</m:mi>
												<m:mo id="id67742">)</m:mo>
												<m:mo id="id67745">⊕</m:mo>
												<m:mo id="id67747">(</m:mo>
												<m:mi id="id67749">Z</m:mi>
												<m:msub id="id67751">
													<m:mo id="id67752">×</m:mo>
													<m:mi id="id67755">T</m:mi>
												</m:msub>
												<m:msup id="id67757">
													<m:mi id="id67758" mathvariant="fraktur">v</m:mi>
													<m:mo id="id67762">*</m:mo>
												</m:msup>
												<m:mo id="id67764">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id67767" encoding="MathML-Content">
												<m:ci id="id67770">(</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
we define <m:math id="id67773" display="inline">
										<m:semantics id="id67777">
											<m:msub id="id67778">
												<m:mi id="id67779">J</m:mi>
												<m:mn id="id67781">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id67783" encoding="MathML-Content">
												<m:apply id="id67786">
													<m:csymbol id="id67788" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67792">J</m:ci>
													<m:cn id="id67794">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to
equal <m:math id="id67798" display="inline">
										<m:semantics id="id67801">
											<m:mfenced id="id67802" open="(" close=")">
												<m:mtable id="id67807" rowspacing="0.2ex" columnspacing="0.4em">
													<m:mtr id="id67813">
														<m:mtd id="id67814" columnalign="center">
															<m:msub id="id67817">
																<m:mi id="id67818">i</m:mi>
																<m:mi id="id67820" mathvariant="fraktur">v</m:mi>
															</m:msub>
														</m:mtd>
														<m:mtd id="id67825" columnalign="center">
															<m:mn id="id67828">0</m:mn>
														</m:mtd>
													</m:mtr>
													<m:mtr id="id67830">
														<m:mtd id="id67831" columnalign="center">
															<m:mn id="id67835">0</m:mn>
														</m:mtd>
														<m:mtd id="id67837" columnalign="center">
															<m:msub id="id67840">
																<m:mi id="id67841">i</m:mi>
																<m:msup id="id67843">
																	<m:mi id="id67844" mathvariant="fraktur">v</m:mi>
																	<m:mo id="id67849">*</m:mo>
																</m:msup>
															</m:msub>
														</m:mtd>
													</m:mtr>
												</m:mtable>
											</m:mfenced>
											<m:annotation-xml id="id67851" encoding="MathML-Content">
												<m:mtext id="id67854">iv00iv*</m:mtext>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p20">
								<p id="id67867" style="" class="p">We now show that <m:math id="id67873" display="inline">
										<m:semantics id="id67876">
											<m:msub id="id67877">
												<m:mi id="id67878">J</m:mi>
												<m:mn id="id67881">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id67883" encoding="MathML-Content">
												<m:apply id="id67886">
													<m:csymbol id="id67887" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67892">J</m:ci>
													<m:cn id="id67894">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id67897" display="inline">
										<m:semantics id="id67901">
											<m:msub id="id67902">
												<m:mi id="id67903">J</m:mi>
												<m:mn id="id67905">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id67907" encoding="MathML-Content">
												<m:apply id="id67910">
													<m:csymbol id="id67912" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id67916">J</m:ci>
													<m:cn id="id67918">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> are homotopic through almost
complex structures. Consider the complex linear transformations <m:math id="id67922" display="inline">
										<m:semantics id="id67926">
											<m:mrow id="id67927">
												<m:msub id="id67928">
													<m:mi id="id67929">j</m:mi>
													<m:mn id="id67931">0</m:mn>
												</m:msub>
												<m:mo id="id67933">:=</m:mo>
												<m:mfenced id="id67935" open="(" close=")">
													<m:mtable id="id67940" rowspacing="0.2ex" columnspacing="0.4em">
														<m:mtr id="id67946">
															<m:mtd id="id67947" columnalign="center">
																<m:mn id="id67950">0</m:mn>
															</m:mtd>
															<m:mtd id="id67953" columnalign="center">
																<m:mrow id="id67956">
																	<m:mo id="id67957">-</m:mo>
																	<m:mn id="id67959">1</m:mn>
																</m:mrow>
															</m:mtd>
														</m:mtr>
														<m:mtr id="id67961">
															<m:mtd id="id67962" columnalign="center">
																<m:mn id="id67966">1</m:mn>
															</m:mtd>
															<m:mtd id="id67968" columnalign="center">
																<m:mn id="id67971">0</m:mn>
															</m:mtd>
														</m:mtr>
													</m:mtable>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id67973" encoding="MathML-Content">
												<m:apply id="id67977">
													<m:ci id="id67978">:=</m:ci>
													<m:apply id="id67980">
														<m:csymbol id="id67981" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id67986">j</m:ci>
														<m:cn id="id67988">0</m:cn>
													</m:apply>
													<m:mtext id="id67990">0-110</m:mtext>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and <m:math id="id67994" display="inline">
										<m:semantics id="id67997">
											<m:mrow id="id67998">
												<m:msub id="id67999">
													<m:mi id="id68000">j</m:mi>
													<m:mn id="id68002">1</m:mn>
												</m:msub>
												<m:mo id="id68004">:=</m:mo>
												<m:mfenced id="id68006" open="(" close=")">
													<m:mtable id="id68012" rowspacing="0.2ex" columnspacing="0.4em">
														<m:mtr id="id68017">
															<m:mtd id="id68018" columnalign="center">
																<m:mi id="id68022">i</m:mi>
															</m:mtd>
															<m:mtd id="id68024" columnalign="center">
																<m:mn id="id68027">0</m:mn>
															</m:mtd>
														</m:mtr>
														<m:mtr id="id68029">
															<m:mtd id="id68030" columnalign="center">
																<m:mn id="id68034">0</m:mn>
															</m:mtd>
															<m:mtd id="id68036" columnalign="center">
																<m:mrow id="id68039">
																	<m:mo id="id68040">-</m:mo>
																	<m:mi id="id68042">i</m:mi>
																</m:mrow>
															</m:mtd>
														</m:mtr>
													</m:mtable>
												</m:mfenced>
												<m:mo id="id68044">∈</m:mo>
												<m:mrow id="id68047">
													<m:mo id="id68048">End</m:mo>
													<m:mo id="id68050">⁡</m:mo>
													<m:mfenced id="id68052" open="(" close=")">
														<m:mrow id="id68057">
															<m:mi id="id68058" mathvariant="double-struck">C</m:mi>
															<m:mo id="id68063">⊕</m:mo>
															<m:mi id="id68065" mathvariant="double-struck">C</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id68070" encoding="MathML-Content">
												<m:apply id="id68073">
													<m:ci id="id68074"/>
													<m:apply id="id68075">
														<m:csymbol id="id68076" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id68081">j</m:ci>
														<m:cn id="id68083">1</m:cn>
													</m:apply>
													<m:ci id="id68085">:=</m:ci>
													<m:mtext id="id68087">i00-i</m:mtext>
													<m:ci id="id68090">∈</m:ci>
													<m:apply id="id68092">
														<m:ci id="id68093">End</m:ci>
														<m:apply id="id68095">
															<m:ci id="id68096">⊕</m:ci>
															<m:ci id="id68099">C</m:ci>
															<m:ci id="id68101">C</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Since <m:math id="id68104" display="inline">
										<m:semantics id="id68108">
											<m:msub id="id68109">
												<m:mi id="id68110">j</m:mi>
												<m:mn id="id68112">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id68114" encoding="MathML-Content">
												<m:apply id="id68117">
													<m:csymbol id="id68118" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68123">j</m:ci>
													<m:cn id="id68125">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id68129" display="inline">
										<m:semantics id="id68132">
											<m:msub id="id68133">
												<m:mi id="id68134">j</m:mi>
												<m:mn id="id68136">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id68138" encoding="MathML-Content">
												<m:apply id="id68142">
													<m:csymbol id="id68143" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68148">j</m:ci>
													<m:cn id="id68150">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
are unitary matrices having the same
eigenvalues, there is a unitary matrix <m:math id="id68154" display="inline">
										<m:semantics id="id68157">
											<m:msub id="id68158">
												<m:mi id="id68159">g</m:mi>
												<m:mn id="id68161">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id68163" encoding="MathML-Content">
												<m:apply id="id68167">
													<m:csymbol id="id68168" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68172">g</m:ci>
													<m:cn id="id68175">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> such that <m:math id="id68178" display="inline">
										<m:semantics id="id68181">
											<m:mrow id="id68182">
												<m:msub id="id68184">
													<m:mi id="id68185">j</m:mi>
													<m:mn id="id68187">1</m:mn>
												</m:msub>
												<m:mo id="id68189">=</m:mo>
												<m:mrow id="id68191">
													<m:msub id="id68192">
														<m:mi id="id68193">g</m:mi>
														<m:mn id="id68195">1</m:mn>
													</m:msub>
													<m:mo id="id68197">⁢</m:mo>
													<m:msub id="id68200">
														<m:mi id="id68201">j</m:mi>
														<m:mn id="id68203">0</m:mn>
													</m:msub>
													<m:mo id="id68205">⁢</m:mo>
													<m:msubsup id="id68208">
														<m:mi id="id68209">g</m:mi>
														<m:mn id="id68211">1</m:mn>
														<m:mrow id="id68213">
															<m:mo id="id68214">-</m:mo>
															<m:mn id="id68216">1</m:mn>
														</m:mrow>
													</m:msubsup>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id68218" encoding="MathML-Content">
												<m:apply id="id68222">
													<m:eq id="id68223"/>
													<m:apply id="id68224">
														<m:csymbol id="id68225" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id68229">j</m:ci>
														<m:cn id="id68232">1</m:cn>
													</m:apply>
													<m:apply id="id68234">
														<m:times id="id68235"/>
														<m:apply id="id68236">
															<m:csymbol id="id68237" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id68242">g</m:ci>
															<m:cn id="id68244">1</m:cn>
														</m:apply>
														<m:apply id="id68246">
															<m:csymbol id="id68247" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id68252">j</m:ci>
															<m:cn id="id68254">0</m:cn>
														</m:apply>
														<m:apply id="id68256">
															<m:csymbol id="id68257" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id68262">
																<m:csymbol id="id68263" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id68267">g</m:ci>
																<m:cn id="id68269">1</m:cn>
															</m:apply>
															<m:apply id="id68272">
																<m:minus id="id68273"/>
																<m:cn id="id68274">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Let <m:math id="id68277" display="inline">
										<m:semantics id="id68280">
											<m:msub id="id68282">
												<m:mi id="id68283">g</m:mi>
												<m:mi id="id68285">t</m:mi>
											</m:msub>
											<m:annotation-xml id="id68287" encoding="MathML-Content">
												<m:apply id="id68290">
													<m:csymbol id="id68291" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68296">g</m:ci>
													<m:ci id="id68298">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, for <m:math id="id68302" display="inline">
										<m:semantics id="id68305">
											<m:mrow id="id68306">
												<m:mi id="id68307">t</m:mi>
												<m:mo id="id68309">∈</m:mo>
												<m:mfenced id="id68312" open="[" close="]">
													<m:mrow id="id68317">
														<m:mn id="id68318">0</m:mn>
														<m:mo id="id68320">,</m:mo>
														<m:mn id="id68322">1</m:mn>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id68324" encoding="MathML-Content">
												<m:apply id="id68327">
													<m:ci id="id68328">∈</m:ci>
													<m:ci id="id68331">t</m:ci>
													<m:apply id="id68333">
														<m:interval id="id68334" closure="closed"/>
														<m:cn id="id68337">0</m:cn>
														<m:cn id="id68340">1</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, be a path of unitary matrices
with <m:math id="id68343" display="inline">
										<m:semantics id="id68346">
											<m:msub id="id68348">
												<m:mi id="id68349">g</m:mi>
												<m:mn id="id68351">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id68353" encoding="MathML-Content">
												<m:apply id="id68356">
													<m:csymbol id="id68357" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68362">g</m:ci>
													<m:cn id="id68364">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> equal to the identity and <m:math id="id68368" display="inline">
										<m:semantics id="id68371">
											<m:msub id="id68372">
												<m:mi id="id68373">g</m:mi>
												<m:mn id="id68375">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id68377" encoding="MathML-Content">
												<m:apply id="id68381">
													<m:csymbol id="id68382" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68386">g</m:ci>
													<m:cn id="id68389">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> the matrix we have just described.
Then <m:math id="id68392" display="inline">
										<m:semantics id="id68396">
											<m:mrow id="id68397">
												<m:msub id="id68398">
													<m:mi id="id68399">j</m:mi>
													<m:mi id="id68401">t</m:mi>
												</m:msub>
												<m:mo id="id68403">:=</m:mo>
												<m:mrow id="id68405">
													<m:msub id="id68406">
														<m:mi id="id68407">g</m:mi>
														<m:mi id="id68410">t</m:mi>
													</m:msub>
													<m:mo id="id68412">⁢</m:mo>
													<m:msub id="id68414">
														<m:mi id="id68415">j</m:mi>
														<m:mn id="id68417">0</m:mn>
													</m:msub>
													<m:mo id="id68419">⁢</m:mo>
													<m:msubsup id="id68422">
														<m:mi id="id68423">g</m:mi>
														<m:mi id="id68425">t</m:mi>
														<m:mrow id="id68427">
															<m:mo id="id68428">-</m:mo>
															<m:mn id="id68430">1</m:mn>
														</m:mrow>
													</m:msubsup>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id68432" encoding="MathML-Content">
												<m:apply id="id68436">
													<m:ci id="id68437">:=</m:ci>
													<m:apply id="id68439">
														<m:csymbol id="id68440" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id68445">j</m:ci>
														<m:ci id="id68447">t</m:ci>
													</m:apply>
													<m:apply id="id68449">
														<m:times id="id68450"/>
														<m:apply id="id68451">
															<m:csymbol id="id68452" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id68457">g</m:ci>
															<m:ci id="id68459">t</m:ci>
														</m:apply>
														<m:apply id="id68461">
															<m:csymbol id="id68462" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id68467">j</m:ci>
															<m:cn id="id68469">0</m:cn>
														</m:apply>
														<m:apply id="id68471">
															<m:csymbol id="id68472" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id68477">
																<m:csymbol id="id68478" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id68483">g</m:ci>
																<m:ci id="id68485">t</m:ci>
															</m:apply>
															<m:apply id="id68487">
																<m:minus id="id68488"/>
																<m:cn id="id68489">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a path of complex structures on the
real vector space underlying <m:math id="id68493" display="inline">
										<m:semantics id="id68496">
											<m:mrow id="id68497">
												<m:mi id="id68498" mathvariant="double-struck">C</m:mi>
												<m:mo id="id68503">⊕</m:mo>
												<m:mi id="id68505" mathvariant="double-struck">C</m:mi>
											</m:mrow>
											<m:annotation-xml id="id68510" encoding="MathML-Content">
												<m:apply id="id68513">
													<m:ci id="id68514">⊕</m:ci>
													<m:ci id="id68516">C</m:ci>
													<m:ci id="id68518">C</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p21">
								<p id="id68530" style="" class="p">Tensoring with <m:math id="id68537" display="inline">
										<m:semantics id="id68540">
											<m:mi id="id68541" mathvariant="fraktur">v</m:mi>
											<m:annotation-xml id="id68546" encoding="MathML-Content">
												<m:ci id="id68549">v</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and using the isomorphism provided by <m:math id="id68553" display="inline">
										<m:semantics id="id68556">
											<m:mi id="id68557">φ</m:mi>
											<m:annotation-xml id="id68560" encoding="MathML-Content">
												<m:ci id="id68563">φ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, we
can thus define a path of almost complex structures <m:math id="id68567" display="inline">
										<m:semantics id="id68570">
											<m:msub id="id68571">
												<m:mi id="id68572">J</m:mi>
												<m:mi id="id68575">t</m:mi>
											</m:msub>
											<m:annotation-xml id="id68577" encoding="MathML-Content">
												<m:apply id="id68580">
													<m:csymbol id="id68581" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68586">J</m:ci>
													<m:ci id="id68588">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> from
									<m:math id="id68591" display="inline">
										<m:semantics id="id68595">
											<m:msub id="id68596">
												<m:mi id="id68597">J</m:mi>
												<m:mn id="id68599">0</m:mn>
											</m:msub>
											<m:annotation-xml id="id68601" encoding="MathML-Content">
												<m:apply id="id68604">
													<m:csymbol id="id68606" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68610">J</m:ci>
													<m:cn id="id68612">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to <m:math id="id68616" display="inline">
										<m:semantics id="id68619">
											<m:msub id="id68620">
												<m:mi id="id68621">J</m:mi>
												<m:mn id="id68623">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id68626" encoding="MathML-Content">
												<m:apply id="id68629">
													<m:csymbol id="id68630" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68635">J</m:ci>
													<m:cn id="id68637">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (keeping the almost complex structure on the subbundle
									<m:math id="id68641" display="inline">
										<m:semantics id="id68644">
											<m:mrow id="id68645">
												<m:msup id="id68646">
													<m:mi id="id68647">π</m:mi>
													<m:mo id="id68650">*</m:mo>
												</m:msup>
												<m:mi id="id68652">T</m:mi>
												<m:mo id="id68654">(</m:mo>
												<m:mi id="id68656">X</m:mi>
												<m:mo id="id68658">/</m:mo>
												<m:mo id="id68660">/</m:mo>
												<m:mi id="id68662">G</m:mi>
												<m:mo id="id68664">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id68667" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> fixed throughout).</p>
							</div>

							<div class="para" id="S1.p22">
								<p id="id68680" style="" class="p">We can think of an almost complex structure over <m:math id="id68687" display="inline">
										<m:semantics id="id68690">
											<m:mrow id="id68691">
												<m:mi id="id68692">X</m:mi>
												<m:mo id="id68694">/</m:mo>
												<m:mo id="id68696">/</m:mo>
												<m:mi id="id68699">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id68701" encoding="MathML-Content">
												<m:ci id="id68704">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> as a section
of a bundle with fibres <m:math id="id68708" display="inline">
										<m:semantics id="id68711">
											<m:mrow id="id68712">
												<m:mrow id="id68713">
													<m:mrow id="id68714">
														<m:mi id="id68715">O</m:mi>
														<m:mo id="id68717">⁢</m:mo>
														<m:mfenced id="id68720" open="(" close=")">
															<m:mrow id="id68725">
																<m:mn id="id68726">2</m:mn>
																<m:mo id="id68728">⁢</m:mo>
																<m:mi id="id68730">n</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id68733">/</m:mo>
													<m:mi id="id68735">U</m:mi>
												</m:mrow>
												<m:mo id="id68737">⁢</m:mo>
												<m:mfenced id="id68739" open="(" close=")">
													<m:mi id="id68744">n</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id68746" encoding="MathML-Content">
												<m:apply id="id68750">
													<m:times id="id68751"/>
													<m:apply id="id68752">
														<m:divide id="id68753"/>
														<m:apply id="id68754">
															<m:times id="id68755"/>
															<m:ci id="id68756">O</m:ci>
															<m:apply id="id68758">
																<m:times id="id68759"/>
																<m:cn id="id68760">2</m:cn>
																<m:ci id="id68763">n</m:ci>
															</m:apply>
														</m:apply>
														<m:ci id="id68765">U</m:ci>
													</m:apply>
													<m:ci id="id68767">n</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, where <m:math id="id68770" display="inline">
										<m:semantics id="id68774">
											<m:mrow id="id68775">
												<m:mn id="id68776">2</m:mn>
												<m:mi id="id68778">n</m:mi>
												<m:mo id="id68780">=</m:mo>
												<m:mo id="id68782">dim</m:mo>
												<m:mi id="id68784">X</m:mi>
												<m:mo id="id68786">/</m:mo>
												<m:mo id="id68789">/</m:mo>
												<m:mi id="id68791">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id68793" encoding="MathML-Content">
												<m:cn id="id68796">2</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> <cite id="id68800" class="cite">[<a id="id68804" href="#bib.bib17" title="" class="ref">17</a>, proposition 2.46]</cite>. Thus <m:math id="id68815" display="inline">
										<m:semantics id="id68818">
											<m:mover id="id68820" accent="true">
												<m:msub id="id68823">
													<m:mi id="id68824">J</m:mi>
													<m:mn id="id68826">0</m:mn>
												</m:msub>
												<m:mo id="id68828">~</m:mo>
											</m:mover>
											<m:annotation-xml id="id68830" encoding="MathML-Content">
												<m:apply id="id68834">
													<m:ci id="id68835">~</m:ci>
													<m:apply id="id68837">
														<m:csymbol id="id68838" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id68843">J</m:ci>
														<m:cn id="id68845">0</m:cn>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is such a
section, and <m:math id="id68848" display="inline">
										<m:semantics id="id68852">
											<m:msub id="id68853">
												<m:mi id="id68854">J</m:mi>
												<m:mi id="id68856">t</m:mi>
											</m:msub>
											<m:annotation-xml id="id68858" encoding="MathML-Content">
												<m:apply id="id68861">
													<m:csymbol id="id68862" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68867">J</m:ci>
													<m:ci id="id68869">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a homotopy of sections restricted to the
submanifold <m:math id="id68873" display="inline">
										<m:semantics id="id68876">
											<m:mrow id="id68878">
												<m:mi id="id68879">Z</m:mi>
												<m:mo id="id68881">/</m:mo>
												<m:mi id="id68883">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id68885" encoding="MathML-Content">
												<m:apply id="id68888">
													<m:divide id="id68889"/>
													<m:ci id="id68890">Z</m:ci>
													<m:ci id="id68893">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. By the homotopy extension property, we can extend
									<m:math id="id68896" display="inline">
										<m:semantics id="id68900">
											<m:msub id="id68901">
												<m:mi id="id68902">J</m:mi>
												<m:mi id="id68904">t</m:mi>
											</m:msub>
											<m:annotation-xml id="id68906" encoding="MathML-Content">
												<m:apply id="id68909">
													<m:csymbol id="id68910" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id68915">J</m:ci>
													<m:ci id="id68917">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to a homotopy <m:math id="id68921" display="inline">
										<m:semantics id="id68924">
											<m:msub id="id68925">
												<m:mover id="id68926" accent="true">
													<m:mi id="id68930">J</m:mi>
													<m:mo id="id68932">~</m:mo>
												</m:mover>
												<m:mi id="id68934">t</m:mi>
											</m:msub>
											<m:annotation-xml id="id68936" encoding="MathML-Content">
												<m:apply id="id68939">
													<m:csymbol id="id68940" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id68945">
														<m:ci id="id68946">~</m:ci>
														<m:ci id="id68948">J</m:ci>
													</m:apply>
													<m:ci id="id68950">t</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of almost complex structures on
									<m:math id="id68954" display="inline">
										<m:semantics id="id68957">
											<m:mrow id="id68958">
												<m:mi id="id68959">X</m:mi>
												<m:mo id="id68962">/</m:mo>
												<m:mo id="id68964">/</m:mo>
												<m:mi id="id68966">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id68968" encoding="MathML-Content">
												<m:ci id="id68971">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S1.p23">
								<p id="id68983" style="" class="p">But <m:math id="id68990" display="inline">
										<m:semantics id="id68993">
											<m:msub id="id68994">
												<m:mover id="id68995" accent="true">
													<m:mi id="id68998">J</m:mi>
													<m:mo id="id69001">~</m:mo>
												</m:mover>
												<m:mn id="id69003">1</m:mn>
											</m:msub>
											<m:annotation-xml id="id69005" encoding="MathML-Content">
												<m:apply id="id69008">
													<m:csymbol id="id69009" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id69014">
														<m:ci id="id69015">~</m:ci>
														<m:ci id="id69017">J</m:ci>
													</m:apply>
													<m:cn id="id69019">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> has the property that <m:math id="id69023" display="inline">
										<m:semantics id="id69026">
											<m:mrow id="id69027">
												<m:mi id="id69028">Z</m:mi>
												<m:mo id="id69030">/</m:mo>
												<m:mi id="id69032">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id69035" encoding="MathML-Content">
												<m:apply id="id69038">
													<m:divide id="id69039"/>
													<m:ci id="id69040">Z</m:ci>
													<m:ci id="id69042">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is an almost complex
submanifold, such that the  complex structure on <m:math id="id69046" display="inline">
										<m:semantics id="id69049">
											<m:mrow id="id69050">
												<m:mi id="id69052">Z</m:mi>
												<m:mo id="id69054">/</m:mo>
												<m:mi id="id69056">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id69058" encoding="MathML-Content">
												<m:apply id="id69061">
													<m:divide id="id69062"/>
													<m:ci id="id69063">Z</m:ci>
													<m:ci id="id69066">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and on its
normal bundle agree with the complex structures on the vector bundles
in the proposition, hence completing the proof.
∎</p>
							</div>

						</div>

					</div>

					<div class="section" id="S2">
						<h2 id="id69082" class="title section-title">
							<span id="id69085" class="tag">2 </span>The integration formula</h2>
						<div class="para" id="S2.p1">
							<p id="id69098" style="" class="p">In this section we prove the integration formula, theorem B. We begin
by recalling some cohomological techniques needed in the proof.</p>
						</div>

						<div class="subsection" id="S2.SS1">
							<h3 id="id69116" class="title subsection-title">
								<span id="id69119" class="tag">2.1 </span>Integration over the fibre</h3>
							<div class="para" id="S2.SS1.p1">
								<p id="id69133" style="" class="p">If <m:math id="id69139" display="inline">
										<m:semantics id="id69142">
											<m:mrow id="id69143">
												<m:mi id="id69144">π</m:mi>
												<m:mo id="id69147">:</m:mo>
												<m:mrow id="id69149">
													<m:mi id="id69150">Y</m:mi>
													<m:mo id="id69152">→</m:mo>
													<m:mi id="id69154">B</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id69156" encoding="MathML-Content">
												<m:apply id="id69160">
													<m:ci id="id69161">:</m:ci>
													<m:ci id="id69163">π</m:ci>
													<m:apply id="id69165">
														<m:ci id="id69166">→</m:ci>
														<m:ci id="id69169">Y</m:ci>
														<m:ci id="id69171">B</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a fibre bundle with fibre <m:math id="id69175" display="inline">
										<m:semantics id="id69178">
											<m:mi id="id69179">F</m:mi>
											<m:annotation-xml id="id69181" encoding="MathML-Content">
												<m:ci id="id69184">F</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, such that <m:math id="id69188" display="inline">
										<m:semantics id="id69191">
											<m:mrow id="id69192">
												<m:mi id="id69193">Y</m:mi>
												<m:mo id="id69196">,</m:mo>
												<m:mi id="id69198">B</m:mi>
											</m:mrow>
											<m:annotation-xml id="id69200" encoding="MathML-Content">
												<m:apply id="id69203">
													<m:list id="id69204"/>
													<m:ci id="id69205">Y</m:ci>
													<m:ci id="id69207">B</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and
									<m:math id="id69211" display="inline">
										<m:semantics id="id69214">
											<m:mi id="id69215">F</m:mi>
											<m:annotation-xml id="id69217" encoding="MathML-Content">
												<m:ci id="id69221">F</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> are  compact oriented manifolds, then <span id="id69224" style="" class="text bold">integration over the
fibre</span> is a map</p>
								<table class="equation" id="S2.Ex1">

									<tr valign="baseline" class="equation" id="S2.Ex1">
										<td id="id69249" class="eqpad"/>
										<td id="id69254" nowrap="yes" align="center" colspan="1">
											<m:math id="id69261" display="block">
												<m:semantics id="id69264">
													<m:mrow id="id69265">
														<m:msub id="id69266">
															<m:mi id="id69267">π</m:mi>
															<m:mo id="id69270">*</m:mo>
														</m:msub>
														<m:mo id="id69272">:</m:mo>
														<m:mrow id="id69274">
															<m:mrow id="id69275">
																<m:msup id="id69276">
																	<m:mi id="id69277">H</m:mi>
																	<m:mo id="id69279">*</m:mo>
																</m:msup>
																<m:mo id="id69282">⁢</m:mo>
																<m:mfenced id="id69284" open="(" close=")">
																	<m:mi id="id69289">Y</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id69291">→</m:mo>
															<m:mrow id="id69294">
																<m:msup id="id69295">
																	<m:mi id="id69296">H</m:mi>
																	<m:mrow id="id69298">
																		<m:none id="id69299"/>
																		<m:mo id="id69300">*</m:mo>
																		<m:mrow id="id69302">
																			<m:mo id="id69303">-</m:mo>
																			<m:mrow id="id69305">
																				<m:mo id="id69306">dim</m:mo>
																				<m:mo id="id69308">⁡</m:mo>
																				<m:mi id="id69311">F</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:mrow>
																</m:msup>
																<m:mo id="id69313">⁢</m:mo>
																<m:mfenced id="id69315" open="(" close=")">
																	<m:mi id="id69320">B</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id69323" encoding="MathML-Content">
														<m:apply id="id69326">
															<m:ci id="id69327">:</m:ci>
															<m:apply id="id69329">
																<m:csymbol id="id69330" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id69335">π</m:ci>
																<m:times id="id69337"/>
															</m:apply>
															<m:apply id="id69338">
																<m:ci id="id69339">→</m:ci>
																<m:apply id="id69342">
																	<m:times id="id69343"/>
																	<m:apply id="id69344">
																		<m:csymbol id="id69345" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id69350">H</m:ci>
																		<m:times id="id69352"/>
																	</m:apply>
																	<m:ci id="id69353">Y</m:ci>
																</m:apply>
																<m:apply id="id69355">
																	<m:times id="id69356"/>
																	<m:apply id="id69357">
																		<m:csymbol id="id69358" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id69363">H</m:ci>
																		<m:apply id="id69365">
																			<m:times id="id69366"/>
																			<m:ci id="id69367"/>
																			<m:apply id="id69368">
																				<m:minus id="id69369"/>
																				<m:apply id="id69370">
																					<m:ci id="id69371">dim</m:ci>
																					<m:ci id="id69374">F</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci id="id69376">B</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id69379" class="eqpad"/>
									</tr>

								</table>
								<p id="id69385" style="" class="p">satisfying</p>

								<dl class="description" id="I3">

									<dt class="item" id="I3.i1">
										<span id="id69405" class="tag">
											<span id="id69408" style="" class="text bold">(Multiplication)</span>
										</span>
									</dt>
									<dd id="id69416" class="item">
										<div class="para" id="I3.i1.p1">
											<p id="id69427" style="" class="p">
												<m:math id="id69432" display="inline">
													<m:semantics id="id69435">
														<m:mrow id="id69436">
															<m:msub id="id69437">
																<m:mi id="id69438">π</m:mi>
																<m:mo id="id69441">*</m:mo>
															</m:msub>
															<m:mo id="id69443">(</m:mo>
															<m:msup id="id69445">
																<m:mi id="id69446">π</m:mi>
																<m:mo id="id69448">*</m:mo>
															</m:msup>
															<m:mo id="id69451">(</m:mo>
															<m:mi id="id69453">b</m:mi>
															<m:mo id="id69455">)</m:mo>
															<m:mo id="id69457">⌣</m:mo>
															<m:mi id="id69459">a</m:mi>
															<m:mo id="id69462">)</m:mo>
															<m:mo id="id69464">=</m:mo>
															<m:mi id="id69466">b</m:mi>
															<m:mo id="id69468">⌣</m:mo>
															<m:msub id="id69470">
																<m:mi id="id69471">π</m:mi>
																<m:mo id="id69474">*</m:mo>
															</m:msub>
															<m:mi id="id69476">a</m:mi>
															<m:mo id="id69478">,</m:mo>
															<m:mi id="id69480" mathvariant="normal">∀</m:mi>
															<m:mi id="id69485">a</m:mi>
															<m:mo id="id69487">∈</m:mo>
															<m:msup id="id69489">
																<m:mi id="id69490">H</m:mi>
																<m:mo id="id69493">*</m:mo>
															</m:msup>
															<m:mo id="id69495">(</m:mo>
															<m:mi id="id69497">Y</m:mi>
															<m:mo id="id69499">)</m:mo>
															<m:mo id="id69501">,</m:mo>
															<m:mi id="id69503">b</m:mi>
															<m:mo id="id69505">∈</m:mo>
															<m:msup id="id69508">
																<m:mi id="id69509">H</m:mi>
																<m:mo id="id69511">*</m:mo>
															</m:msup>
															<m:mo id="id69513">(</m:mo>
															<m:mi id="id69515">B</m:mi>
															<m:mo id="id69517">)</m:mo>
															<m:mo id="id69520">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id69522" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</p>
										</div>
									</dd>

									<dt class="item" id="I3.i2">
										<span id="id69534" class="tag">
											<span id="id69537" style="" class="text bold">(Restriction)</span>
										</span>
									</dt>
									<dd id="id69545" class="item">
										<div class="para" id="I3.i2.p1">
											<p id="id69556" style="" class="p">If <m:math id="id69562" display="inline">
													<m:semantics id="id69565">
														<m:mrow id="id69566">
															<m:mi id="id69567">i</m:mi>
															<m:mo id="id69569">:</m:mo>
															<m:mrow id="id69572">
																<m:mi id="id69573">S</m:mi>
																<m:mo id="id69575">↪</m:mo>
																<m:mi id="id69577">B</m:mi>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id69579" encoding="MathML-Content">
															<m:apply id="id69583">
																<m:ci id="id69584">:</m:ci>
																<m:ci id="id69586">i</m:ci>
																<m:apply id="id69588">
																	<m:ci id="id69589">↪</m:ci>
																	<m:ci id="id69591">S</m:ci>
																	<m:ci id="id69594">B</m:ci>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> is the inclusion a closed oriented
submanifold, then the following square of maps commutes</p>
											<table class="equation" id="S2.Ex2">

												<tr valign="baseline" class="equation" id="S2.Ex2">
													<td id="id69614" class="eqpad"/>
													<td id="id69619" nowrap="yes" align="center" colspan="1">
														<m:math id="id69626" display="block">
															<m:semantics id="id69629">
																<m:mrow id="id69630">
																	<m:mrow id="id69632">
																		<m:msup id="id69633">
																			<m:mi id="id69634">H</m:mi>
																			<m:mo id="id69636">*</m:mo>
																		</m:msup>
																		<m:mo id="id69638">⁢</m:mo>
																		<m:mfenced id="id69640" open="(" close=")">
																			<m:mrow id="id69645">
																				<m:msup id="id69646">
																					<m:mi id="id69648">π</m:mi>
																					<m:mrow id="id69650">
																						<m:mo id="id69651">-</m:mo>
																						<m:mn id="id69653">1</m:mn>
																					</m:mrow>
																				</m:msup>
																				<m:mo id="id69655">⁢</m:mo>
																				<m:mfenced id="id69658" open="(" close=")">
																					<m:mi id="id69663">S</m:mi>
																				</m:mfenced>
																			</m:mrow>
																		</m:mfenced>
																		<m:mo id="id69665">⁢</m:mo>
																		<m:msup id="id69667">
																			<m:mfenced id="id69668" open="[" close="]">
																				<m:mi id="id69673">d</m:mi>
																			</m:mfenced>
																			<m:msubsup id="id69676">
																				<m:mi id="id69677">π</m:mi>
																				<m:mo id="id69679">*</m:mo>
																				<m:mo id="id69681">′</m:mo>
																			</m:msubsup>
																		</m:msup>
																		<m:mo id="id69684">⁢</m:mo>
																		<m:msup id="id69686">
																			<m:mi id="id69687">H</m:mi>
																			<m:mo id="id69689">*</m:mo>
																		</m:msup>
																		<m:mo id="id69691">⁢</m:mo>
																		<m:mfenced id="id69694" open="(" close=")">
																			<m:mi id="id69699">Y</m:mi>
																		</m:mfenced>
																		<m:mo id="id69701">⁢</m:mo>
																		<m:msup id="id69703">
																			<m:mfenced id="id69704" open="[" close="]">
																				<m:mi id="id69709">l</m:mi>
																			</m:mfenced>
																			<m:mrow id="id69712">
																				<m:msup id="id69713">
																					<m:mi id="id69714">i</m:mi>
																					<m:mo id="id69716">′</m:mo>
																				</m:msup>
																				<m:msup id="id69718">
																					<m:mi id="id69719"/>
																					<m:mo id="id69720">*</m:mo>
																				</m:msup>
																			</m:mrow>
																		</m:msup>
																		<m:mo id="id69722">⁢</m:mo>
																		<m:msup id="id69725">
																			<m:mfenced id="id69726" open="[" close="]">
																				<m:mi id="id69731">d</m:mi>
																			</m:mfenced>
																			<m:msub id="id69733">
																				<m:mi id="id69734">π</m:mi>
																				<m:mo id="id69737">*</m:mo>
																			</m:msub>
																		</m:msup>
																		<m:mo id="id69739">⁢</m:mo>
																		<m:msup id="id69741">
																			<m:mi id="id69742">H</m:mi>
																			<m:mrow id="id69744">
																				<m:none id="id69745"/>
																				<m:mo id="id69746">*</m:mo>
																				<m:mrow id="id69749">
																					<m:mo id="id69750">-</m:mo>
																					<m:mrow id="id69752">
																						<m:mo id="id69753">dim</m:mo>
																						<m:mo id="id69755">⁡</m:mo>
																						<m:mi id="id69757">F</m:mi>
																					</m:mrow>
																				</m:mrow>
																			</m:mrow>
																		</m:msup>
																		<m:mo id="id69760">⁢</m:mo>
																		<m:mfenced id="id69762" open="(" close=")">
																			<m:mi id="id69767">S</m:mi>
																		</m:mfenced>
																		<m:mo id="id69769">⁢</m:mo>
																		<m:msup id="id69772">
																			<m:mi id="id69773">H</m:mi>
																			<m:mrow id="id69775">
																				<m:none id="id69776"/>
																				<m:mo id="id69777">*</m:mo>
																				<m:mrow id="id69779">
																					<m:mo id="id69780">-</m:mo>
																					<m:mrow id="id69782">
																						<m:mo id="id69783">dim</m:mo>
																						<m:mo id="id69785">⁡</m:mo>
																						<m:mi id="id69788">F</m:mi>
																					</m:mrow>
																				</m:mrow>
																			</m:mrow>
																		</m:msup>
																		<m:mo id="id69790">⁢</m:mo>
																		<m:mfenced id="id69792" open="(" close=")">
																			<m:mi id="id69797">B</m:mi>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo id="id69800">,</m:mo>
																	<m:msub id="id69802">
																		<m:mfenced id="id69803" open="[" close="]">
																			<m:mi id="id69808">l</m:mi>
																		</m:mfenced>
																		<m:msup id="id69810">
																			<m:mi id="id69811">i</m:mi>
																			<m:mo id="id69813">*</m:mo>
																		</m:msup>
																	</m:msub>
																</m:mrow>
																<m:annotation-xml id="id69815" encoding="MathML-Content">
																	<m:apply id="id69819">
																		<m:list id="id69820"/>
																		<m:apply id="id69821">
																			<m:times id="id69822"/>
																			<m:apply id="id69823">
																				<m:csymbol id="id69824" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69829">H</m:ci>
																				<m:times id="id69831"/>
																			</m:apply>
																			<m:apply id="id69832">
																				<m:times id="id69833"/>
																				<m:apply id="id69834">
																					<m:csymbol id="id69835" cd="ambiguous">superscript</m:csymbol>
																					<m:ci id="id69840">π</m:ci>
																					<m:apply id="id69842">
																						<m:minus id="id69843"/>
																						<m:cn id="id69844">1</m:cn>
																					</m:apply>
																				</m:apply>
																				<m:ci id="id69846">S</m:ci>
																			</m:apply>
																			<m:apply id="id69848">
																				<m:csymbol id="id69850" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69854">d</m:ci>
																				<m:apply id="id69856">
																					<m:csymbol id="id69857" cd="ambiguous">subscript</m:csymbol>
																					<m:apply id="id69862">
																						<m:csymbol id="id69863" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id69868">π</m:ci>
																						<m:ci id="id69870">′</m:ci>
																					</m:apply>
																					<m:times id="id69873"/>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id69874">
																				<m:csymbol id="id69875" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69879">H</m:ci>
																				<m:times id="id69882"/>
																			</m:apply>
																			<m:ci id="id69883">Y</m:ci>
																			<m:apply id="id69885">
																				<m:csymbol id="id69886" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69890">l</m:ci>
																				<m:mtext id="id69893">i′*</m:mtext>
																			</m:apply>
																			<m:apply id="id69895">
																				<m:csymbol id="id69896" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69901">d</m:ci>
																				<m:apply id="id69903">
																					<m:csymbol id="id69904" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id69909">π</m:ci>
																					<m:times id="id69911"/>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id69912">
																				<m:csymbol id="id69913" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69918">H</m:ci>
																				<m:apply id="id69920">
																					<m:times id="id69921"/>
																					<m:ci id="id69922"/>
																					<m:apply id="id69923">
																						<m:minus id="id69924"/>
																						<m:apply id="id69925">
																							<m:ci id="id69926">dim</m:ci>
																							<m:ci id="id69928">F</m:ci>
																						</m:apply>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id69931">S</m:ci>
																			<m:apply id="id69933">
																				<m:csymbol id="id69934" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69938">H</m:ci>
																				<m:apply id="id69941">
																					<m:times id="id69942"/>
																					<m:ci id="id69943"/>
																					<m:apply id="id69944">
																						<m:minus id="id69945"/>
																						<m:apply id="id69946">
																							<m:ci id="id69947">dim</m:ci>
																							<m:ci id="id69949">F</m:ci>
																						</m:apply>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id69951">B</m:ci>
																		</m:apply>
																		<m:apply id="id69953">
																			<m:csymbol id="id69954" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id69959">l</m:ci>
																			<m:apply id="id69961">
																				<m:csymbol id="id69962" cd="ambiguous">superscript</m:csymbol>
																				<m:ci id="id69967">i</m:ci>
																				<m:times id="id69969"/>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:annotation-xml>
															</m:semantics>
														</m:math>
													</td>
													<td id="id69971" class="eqpad"/>
												</tr>

											</table>
											<p id="id69978" style="" class="p">where <m:math id="id69984" display="inline">
													<m:semantics id="id69988">
														<m:msup id="id69989">
															<m:mi id="id69990">π</m:mi>
															<m:mo id="id69992">′</m:mo>
														</m:msup>
														<m:annotation-xml id="id69994" encoding="MathML-Content">
															<m:apply id="id69998">
																<m:csymbol id="id69999" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id70004">π</m:ci>
																<m:ci id="id70006">′</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> denotes the restriction of <m:math id="id70010" display="inline">
													<m:semantics id="id70013">
														<m:mi id="id70014">π</m:mi>
														<m:annotation-xml id="id70017" encoding="MathML-Content">
															<m:ci id="id70020">π</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> to <m:math id="id70024" display="inline">
													<m:semantics id="id70027">
														<m:mrow id="id70028">
															<m:msup id="id70029">
																<m:mi id="id70030">π</m:mi>
																<m:mrow id="id70033">
																	<m:mo id="id70034">-</m:mo>
																	<m:mn id="id70036">1</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo id="id70038">⁢</m:mo>
															<m:mfenced id="id70040" open="(" close=")">
																<m:mi id="id70045">S</m:mi>
															</m:mfenced>
														</m:mrow>
														<m:annotation-xml id="id70048" encoding="MathML-Content">
															<m:apply id="id70051">
																<m:times id="id70052"/>
																<m:apply id="id70053">
																	<m:csymbol id="id70054" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id70059">π</m:ci>
																	<m:apply id="id70061">
																		<m:minus id="id70062"/>
																		<m:cn id="id70063">1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci id="id70065">S</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, and
												<m:math id="id70069" display="inline">
													<m:semantics id="id70072">
														<m:msup id="id70073">
															<m:mi id="id70074">i</m:mi>
															<m:mo id="id70076">′</m:mo>
														</m:msup>
														<m:annotation-xml id="id70079" encoding="MathML-Content">
															<m:apply id="id70082">
																<m:csymbol id="id70083" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id70088">i</m:ci>
																<m:ci id="id70090">′</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> denotes the inclusion of <m:math id="id70094" display="inline">
													<m:semantics id="id70097">
														<m:mrow id="id70098">
															<m:msup id="id70099">
																<m:mi id="id70100">π</m:mi>
																<m:mrow id="id70103">
																	<m:mo id="id70104">-</m:mo>
																	<m:mn id="id70106">1</m:mn>
																</m:mrow>
															</m:msup>
															<m:mo id="id70108">⁢</m:mo>
															<m:mfenced id="id70111" open="(" close=")">
																<m:mi id="id70116">S</m:mi>
															</m:mfenced>
														</m:mrow>
														<m:annotation-xml id="id70118" encoding="MathML-Content">
															<m:apply id="id70121">
																<m:times id="id70122"/>
																<m:apply id="id70123">
																	<m:csymbol id="id70124" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id70129">π</m:ci>
																	<m:apply id="id70131">
																		<m:minus id="id70132"/>
																		<m:cn id="id70134">1</m:cn>
																	</m:apply>
																</m:apply>
																<m:ci id="id70136">S</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> in <m:math id="id70139" display="inline">
													<m:semantics id="id70142">
														<m:mi id="id70144">Y</m:mi>
														<m:annotation-xml id="id70146" encoding="MathML-Content">
															<m:ci id="id70149">Y</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>.</p>
										</div>
									</dd>

									<dt class="item" id="I3.i3">
										<span id="id70161" class="tag">
											<span id="id70164" style="" class="text bold">(Composition)</span>
										</span>
									</dt>
									<dd id="id70172" class="item">
										<div class="para" id="I3.i3.p1">
											<p id="id70183" style="" class="p">
												<m:math id="id70188" display="inline">
													<m:semantics id="id70191">
														<m:mrow id="id70192">
															<m:mrow id="id70193">
																<m:mrow id="id70194">
																	<m:mrow id="id70196">
																		<m:msub id="id70197">
																			<m:mo id="id70198">∫</m:mo>
																			<m:mi id="id70200">B</m:mi>
																		</m:msub>
																		<m:msub id="id70202">
																			<m:mi id="id70203">π</m:mi>
																			<m:mo id="id70206">*</m:mo>
																		</m:msub>
																		<m:mo id="id70208">⁢</m:mo>
																		<m:mi id="id70210">a</m:mi>
																	</m:mrow>
																	<m:mo id="id70212">=</m:mo>
																	<m:mrow id="id70214">
																		<m:msub id="id70216">
																			<m:mo id="id70217">∫</m:mo>
																			<m:mi id="id70219">Y</m:mi>
																		</m:msub>
																		<m:mi id="id70221">a</m:mi>
																	</m:mrow>
																</m:mrow>
																<m:mo id="id70223">,</m:mo>
																<m:mrow id="id70225">
																	<m:mrow id="id70226">
																		<m:mi id="id70228" mathvariant="normal">∀</m:mi>
																		<m:mi id="id70232">a</m:mi>
																	</m:mrow>
																	<m:mo id="id70234">∈</m:mo>
																	<m:mrow id="id70237">
																		<m:msup id="id70238">
																			<m:mi id="id70239">H</m:mi>
																			<m:mo id="id70241">*</m:mo>
																		</m:msup>
																		<m:mo id="id70243">⁢</m:mo>
																		<m:mfenced id="id70246" open="(" close=")">
																			<m:mi id="id70251">Y</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:mrow>
															</m:mrow>
															<m:mo id="id70253">.</m:mo>
														</m:mrow>
														<m:annotation-xml id="id70255" encoding="MathML-Content">
															<m:apply id="id70258">
																<m:ci id="id70259"/>
																<m:apply id="id70260">
																	<m:eq id="id70261"/>
																	<m:apply id="id70262">
																		<m:apply id="id70264">
																			<m:csymbol id="id70265" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id70269"/>
																			<m:ci id="id70270">B</m:ci>
																		</m:apply>
																		<m:apply id="id70272">
																			<m:times id="id70274"/>
																			<m:apply id="id70275">
																				<m:csymbol id="id70276" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id70280">π</m:ci>
																				<m:times id="id70283"/>
																			</m:apply>
																			<m:ci id="id70284">a</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id70286">
																		<m:apply id="id70287">
																			<m:csymbol id="id70288" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id70293"/>
																			<m:ci id="id70294">Y</m:ci>
																		</m:apply>
																		<m:ci id="id70296">a</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id70298">
																	<m:ci id="id70299">∈</m:ci>
																	<m:apply id="id70302">
																		<m:ci id="id70303">∀</m:ci>
																		<m:ci id="id70305">a</m:ci>
																	</m:apply>
																	<m:apply id="id70307">
																		<m:times id="id70308"/>
																		<m:apply id="id70309">
																			<m:csymbol id="id70310" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id70315">H</m:ci>
																			<m:times id="id70317"/>
																		</m:apply>
																		<m:ci id="id70318">Y</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</p>
										</div>
									</dd>

								</dl>
							</div>

						</div>

						<div class="subsection" id="S2.SS2">
							<h3 id="id70333" class="title subsection-title">
								<span id="id70337" class="tag">2.2 </span>The Euler class of a vector bundle</h3>
							<div class="para" id="S2.SS2.p1">
								<p id="id70350" style="" class="p">The second fact we need involves the Euler class of a vector bundle.
Suppose <m:math id="id70357" display="inline">
										<m:semantics id="id70361">
											<m:mi id="id70362">V</m:mi>
											<m:annotation-xml id="id70364" encoding="MathML-Content">
												<m:ci id="id70367">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a real oriented vector bundle
over a compact oriented manifold <m:math id="id70371" display="inline">
										<m:semantics id="id70374">
											<m:mi id="id70375">Y</m:mi>
											<m:annotation-xml id="id70378" encoding="MathML-Content">
												<m:ci id="id70381">Y</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and <m:math id="id70384" display="inline">
										<m:semantics id="id70388">
											<m:mi id="id70389">s</m:mi>
											<m:annotation-xml id="id70391" encoding="MathML-Content">
												<m:ci id="id70394">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a section of <m:math id="id70398" display="inline">
										<m:semantics id="id70401">
											<m:mi id="id70402">V</m:mi>
											<m:annotation-xml id="id70404" encoding="MathML-Content">
												<m:ci id="id70408">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
which is transverse to the zero section. Then the zeroset of <m:math id="id70411" display="inline">
										<m:semantics id="id70415">
											<m:mi id="id70416">s</m:mi>
											<m:annotation-xml id="id70418" encoding="MathML-Content">
												<m:ci id="id70421">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a
submanifold <m:math id="id70425" display="inline">
										<m:semantics id="id70428">
											<m:mi id="id70429">S</m:mi>
											<m:annotation-xml id="id70431" encoding="MathML-Content">
												<m:ci id="id70435">S</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of <m:math id="id70438" display="inline">
										<m:semantics id="id70441">
											<m:mi id="id70442">Y</m:mi>
											<m:annotation-xml id="id70445" encoding="MathML-Content">
												<m:ci id="id70448">Y</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and denoting by <m:math id="id70451" display="inline">
										<m:semantics id="id70455">
											<m:mrow id="id70456">
												<m:mi id="id70457">i</m:mi>
												<m:mo id="id70459">:</m:mo>
												<m:mrow id="id70461">
													<m:mi id="id70462">S</m:mi>
													<m:mo id="id70464">↪</m:mo>
													<m:mi id="id70467">Y</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id70469" encoding="MathML-Content">
												<m:apply id="id70472">
													<m:ci id="id70473">:</m:ci>
													<m:ci id="id70475">i</m:ci>
													<m:apply id="id70478">
														<m:ci id="id70479">↪</m:ci>
														<m:ci id="id70481">S</m:ci>
														<m:ci id="id70483">Y</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> the inclusion, we have</p>
								<table class="equation" id="S2.Ex3">

									<tr valign="baseline" class="equation" id="S2.Ex3">
										<td id="id70503" class="eqpad"/>
										<td id="id70508" nowrap="yes" align="center" colspan="1">
											<m:math id="id70515" display="block">
												<m:semantics id="id70518">
													<m:mrow id="id70519">
														<m:mrow id="id70520">
															<m:mrow id="id70522">
																<m:msub id="id70523">
																	<m:mo id="id70524">∫</m:mo>
																	<m:mi id="id70526">S</m:mi>
																</m:msub>
																<m:msup id="id70528">
																	<m:mi id="id70529">i</m:mi>
																	<m:mo id="id70531">*</m:mo>
																</m:msup>
																<m:mo id="id70534">⁢</m:mo>
																<m:mi id="id70536">a</m:mi>
															</m:mrow>
															<m:mo id="id70538">=</m:mo>
															<m:mrow id="id70540">
																<m:msub id="id70541">
																	<m:mo id="id70542">∫</m:mo>
																	<m:mi id="id70545">Y</m:mi>
																</m:msub>
																<m:mi id="id70547">a</m:mi>
															</m:mrow>
															<m:mo id="id70549">⌣</m:mo>
															<m:mrow id="id70551">
																<m:mi id="id70552">e</m:mi>
																<m:mo id="id70555">⁢</m:mo>
																<m:mfenced id="id70557" open="(" close=")">
																	<m:mi id="id70562">V</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
														<m:mo id="id70564">,</m:mo>
														<m:mrow id="id70566">
															<m:mrow id="id70567">
																<m:mi id="id70568" mathvariant="normal">∀</m:mi>
																<m:mi id="id70573">a</m:mi>
															</m:mrow>
															<m:mo id="id70575">∈</m:mo>
															<m:mrow id="id70578">
																<m:msup id="id70579">
																	<m:mi id="id70580">H</m:mi>
																	<m:mo id="id70582">*</m:mo>
																</m:msup>
																<m:mo id="id70584">⁢</m:mo>
																<m:mfenced id="id70586" open="(" close=")">
																	<m:mi id="id70592">Y</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id70594" encoding="MathML-Content">
														<m:apply id="id70597">
															<m:ci id="id70598"/>
															<m:apply id="id70599">
																<m:ci id="id70600"/>
																<m:apply id="id70601">
																	<m:apply id="id70602">
																		<m:csymbol id="id70603" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id70608"/>
																		<m:ci id="id70609">S</m:ci>
																	</m:apply>
																	<m:apply id="id70611">
																		<m:times id="id70612"/>
																		<m:apply id="id70613">
																			<m:csymbol id="id70614" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id70619">i</m:ci>
																			<m:times id="id70621"/>
																		</m:apply>
																		<m:ci id="id70622">a</m:ci>
																	</m:apply>
																</m:apply>
																<m:eq id="id70624"/>
																<m:apply id="id70626">
																	<m:apply id="id70627">
																		<m:csymbol id="id70628" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id70632"/>
																		<m:ci id="id70633">Y</m:ci>
																	</m:apply>
																	<m:ci id="id70636">a</m:ci>
																</m:apply>
																<m:ci id="id70638">⌣</m:ci>
																<m:apply id="id70640">
																	<m:times id="id70641"/>
																	<m:ci id="id70642">e</m:ci>
																	<m:ci id="id70644">V</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id70646">
																<m:ci id="id70648">∈</m:ci>
																<m:apply id="id70650">
																	<m:ci id="id70651">∀</m:ci>
																	<m:ci id="id70653">a</m:ci>
																</m:apply>
																<m:apply id="id70656">
																	<m:times id="id70657"/>
																	<m:apply id="id70658">
																		<m:csymbol id="id70659" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id70663">H</m:ci>
																		<m:times id="id70666"/>
																	</m:apply>
																	<m:ci id="id70667">Y</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id70670" class="eqpad"/>
									</tr>

								</table>
								<p id="id70676" style="" class="p">where <m:math id="id70683" display="inline">
										<m:semantics id="id70686">
											<m:mrow id="id70687">
												<m:mrow id="id70688">
													<m:mi id="id70689">e</m:mi>
													<m:mo id="id70691">⁢</m:mo>
													<m:mfenced id="id70694" open="(" close=")">
														<m:mi id="id70699">V</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id70701">∈</m:mo>
												<m:mrow id="id70703">
													<m:msup id="id70704">
														<m:mi id="id70706">H</m:mi>
														<m:mrow id="id70708">
															<m:mrow id="id70709">
																<m:mi id="id70710">r</m:mi>
																<m:mo id="id70712">⁢</m:mo>
																<m:mi id="id70714">k</m:mi>
															</m:mrow>
															<m:mo id="id70716">⁢</m:mo>
															<m:mi id="id70719">V</m:mi>
														</m:mrow>
													</m:msup>
													<m:mo id="id70721">⁢</m:mo>
													<m:mfenced id="id70723" open="(" close=")">
														<m:mi id="id70728">Y</m:mi>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id70731" encoding="MathML-Content">
												<m:apply id="id70734">
													<m:ci id="id70735">∈</m:ci>
													<m:apply id="id70737">
														<m:times id="id70738"/>
														<m:ci id="id70740">e</m:ci>
														<m:ci id="id70742">V</m:ci>
													</m:apply>
													<m:apply id="id70744">
														<m:times id="id70745"/>
														<m:apply id="id70746">
															<m:csymbol id="id70747" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id70752">H</m:ci>
															<m:apply id="id70754">
																<m:times id="id70755"/>
																<m:ci id="id70756"/>
																<m:ci id="id70757">V</m:ci>
															</m:apply>
														</m:apply>
														<m:ci id="id70759">Y</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is the Euler class of <m:math id="id70763" display="inline">
										<m:semantics id="id70766">
											<m:mi id="id70767">V</m:mi>
											<m:annotation-xml id="id70769" encoding="MathML-Content">
												<m:ci id="id70772">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. There is a
natural identification <m:math id="id70776" display="inline">
										<m:semantics id="id70779">
											<m:mrow id="id70780">
												<m:mrow id="id70782">
													<m:mi id="id70783">ν</m:mi>
													<m:mo id="id70785">⁢</m:mo>
													<m:mi id="id70787">S</m:mi>
												</m:mrow>
												<m:mo id="id70790">≅</m:mo>
												<m:msub id="id70792">
													<m:mi id="id70793">V</m:mi>
													<m:mi id="id70795">S</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id70797" encoding="MathML-Content">
												<m:apply id="id70801">
													<m:approx id="id70802"/>
													<m:apply id="id70803">
														<m:times id="id70804"/>
														<m:ci id="id70805">ν</m:ci>
														<m:ci id="id70807">S</m:ci>
													</m:apply>
													<m:apply id="id70809">
														<m:csymbol id="id70810" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id70815">V</m:ci>
														<m:ci id="id70817">S</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of the normal bundle to
									<m:math id="id70821" display="inline">
										<m:semantics id="id70824">
											<m:mi id="id70825">S</m:mi>
											<m:annotation-xml id="id70827" encoding="MathML-Content">
												<m:ci id="id70831">S</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> with the restriction of <m:math id="id70834" display="inline">
										<m:semantics id="id70838">
											<m:mi id="id70839">V</m:mi>
											<m:annotation-xml id="id70841" encoding="MathML-Content">
												<m:ci id="id70844">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and hence the orientation of <m:math id="id70848" display="inline">
										<m:semantics id="id70851">
											<m:mi id="id70852">V</m:mi>
											<m:annotation-xml id="id70854" encoding="MathML-Content">
												<m:ci id="id70858">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
induces an orientation on <m:math id="id70861" display="inline">
										<m:semantics id="id70865">
											<m:mrow id="id70866">
												<m:mi id="id70867">ν</m:mi>
												<m:mo id="id70869">⁢</m:mo>
												<m:mi id="id70872">S</m:mi>
											</m:mrow>
											<m:annotation-xml id="id70874" encoding="MathML-Content">
												<m:apply id="id70877">
													<m:times id="id70878"/>
													<m:ci id="id70879">ν</m:ci>
													<m:ci id="id70882">S</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>; we assume <m:math id="id70885" display="inline">
										<m:semantics id="id70888">
											<m:mrow id="id70889">
												<m:mi id="id70890">S</m:mi>
												<m:mo id="id70893">,</m:mo>
												<m:mrow id="id70895">
													<m:mi id="id70896">ν</m:mi>
													<m:mo id="id70898">⁢</m:mo>
													<m:mi id="id70901">S</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id70903" encoding="MathML-Content">
												<m:apply id="id70906">
													<m:list id="id70907"/>
													<m:ci id="id70908">S</m:ci>
													<m:apply id="id70910">
														<m:times id="id70911"/>
														<m:ci id="id70912">ν</m:ci>
														<m:ci id="id70915">S</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id70918" display="inline">
										<m:semantics id="id70922">
											<m:mi id="id70923">Y</m:mi>
											<m:annotation-xml id="id70925" encoding="MathML-Content">
												<m:ci id="id70928">Y</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> are
oriented compatibly.</p>
							</div>

							<div class="para" id="S2.SS2.p2">
								<p id="id70941" style="" class="p">If instead <m:math id="id70947" display="inline">
										<m:semantics id="id70950">
											<m:mi id="id70951">V</m:mi>
											<m:annotation-xml id="id70954" encoding="MathML-Content">
												<m:ci id="id70957">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a complex vector bundle, with its natural real
orientation, then the Euler class of <m:math id="id70961" display="inline">
										<m:semantics id="id70964">
											<m:mi id="id70965">V</m:mi>
											<m:annotation-xml id="id70968" encoding="MathML-Content">
												<m:ci id="id70971">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (thought of as a real vector
bundle) equals the top Chern class of <m:math id="id70975" display="inline">
										<m:semantics id="id70978">
											<m:mi id="id70979">V</m:mi>
											<m:annotation-xml id="id70981" encoding="MathML-Content">
												<m:ci id="id70985">V</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (thought of as a complex
vector bundle). For proofs and more detailed explanations, see Bott
and Tu <cite id="id70990" class="cite">[<a id="id70994" href="#bib.bib4" title="" class="ref">4</a>, section 6]</cite>.</p>
							</div>

						</div>

						<div class="subsection" id="S2.SS3">
							<h3 id="id71015" class="title subsection-title">
								<span id="id71018" class="tag">2.3 </span>Proof of the integration formula</h3>
							<div class="para" id="S2.SS3.p1">
								<p id="id71032" style="" class="p">Recall that we have maps
									<m:math id="id71038" display="inline">
										<m:semantics id="id71042">
											<m:mrow id="id71043">
												<m:mi id="id71044">i</m:mi>
												<m:mo id="id71046">:</m:mo>
												<m:msubsup id="id71048">
													<m:mi id="id71049">μ</m:mi>
													<m:mi id="id71051">G</m:mi>
													<m:mrow id="id71054">
														<m:mo id="id71055">-</m:mo>
														<m:mn id="id71057">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id71059">(</m:mo>
												<m:mn id="id71061">0</m:mn>
												<m:mo id="id71063">)</m:mo>
												<m:mo id="id71065">/</m:mo>
												<m:mi id="id71067">T</m:mi>
												<m:mo id="id71070">↪</m:mo>
												<m:mi id="id71072">X</m:mi>
												<m:mo id="id71074">/</m:mo>
												<m:mo id="id71076">/</m:mo>
												<m:mi id="id71078">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id71080" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> and <m:math id="id71085" display="inline">
										<m:semantics id="id71088">
											<m:mrow id="id71090">
												<m:mi id="id71091">π</m:mi>
												<m:mo id="id71093">:</m:mo>
												<m:msubsup id="id71095">
													<m:mi id="id71096">μ</m:mi>
													<m:mi id="id71099">G</m:mi>
													<m:mrow id="id71101">
														<m:mo id="id71102">-</m:mo>
														<m:mn id="id71104">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id71106">(</m:mo>
												<m:mn id="id71108">0</m:mn>
												<m:mo id="id71110">)</m:mo>
												<m:mo id="id71112">/</m:mo>
												<m:mi id="id71115">T</m:mi>
												<m:mo id="id71117">→</m:mo>
												<m:mi id="id71119">X</m:mi>
												<m:mo id="id71121">/</m:mo>
												<m:mo id="id71123">/</m:mo>
												<m:mi id="id71126">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id71128" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> as defined in section <a id="id71134" href="#S1" title="1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id71133" style="" class="text tag">1</span>
									</a>,
and we say that a class <m:math id="id71149" display="inline">
										<m:semantics id="id71152">
											<m:mrow id="id71153">
												<m:mover id="id71154" accent="true">
													<m:mi id="id71157">a</m:mi>
													<m:mo id="id71160">~</m:mo>
												</m:mover>
												<m:mo id="id71162">∈</m:mo>
												<m:msup id="id71164">
													<m:mi id="id71165">H</m:mi>
													<m:mo id="id71167">*</m:mo>
												</m:msup>
												<m:mo id="id71169">(</m:mo>
												<m:mi id="id71172">X</m:mi>
												<m:mo id="id71174">/</m:mo>
												<m:mo id="id71176">/</m:mo>
												<m:mi id="id71178">T</m:mi>
												<m:mo id="id71180">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id71182" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> is a <span id="id71187" style="" class="text bold">lift</span> of
									<m:math id="id71195" display="inline">
										<m:semantics id="id71198">
											<m:mrow id="id71199">
												<m:mi id="id71200">a</m:mi>
												<m:mo id="id71202">∈</m:mo>
												<m:msup id="id71205">
													<m:mi id="id71206">H</m:mi>
													<m:mo id="id71208">*</m:mo>
												</m:msup>
												<m:mo id="id71210">(</m:mo>
												<m:mi id="id71212">X</m:mi>
												<m:mo id="id71214">/</m:mo>
												<m:mo id="id71217">/</m:mo>
												<m:mi id="id71219">G</m:mi>
												<m:mo id="id71221">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id71223" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> if <m:math id="id71228" display="inline">
										<m:semantics id="id71231">
											<m:mrow id="id71232">
												<m:mrow id="id71233">
													<m:msup id="id71234">
														<m:mi id="id71235">π</m:mi>
														<m:mo id="id71238">*</m:mo>
													</m:msup>
													<m:mo id="id71240">⁢</m:mo>
													<m:mfenced id="id71242" open="(" close=")">
														<m:mi id="id71247">a</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id71249">=</m:mo>
												<m:mrow id="id71252">
													<m:msup id="id71253">
														<m:mi id="id71254">i</m:mi>
														<m:mo id="id71256">*</m:mo>
													</m:msup>
													<m:mo id="id71258">⁢</m:mo>
													<m:mfenced id="id71260" open="(" close=")">
														<m:mover id="id71265" accent="true">
															<m:mi id="id71269">a</m:mi>
															<m:mo id="id71271">~</m:mo>
														</m:mover>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id71273" encoding="MathML-Content">
												<m:apply id="id71276">
													<m:eq id="id71277"/>
													<m:apply id="id71278">
														<m:times id="id71280"/>
														<m:apply id="id71281">
															<m:csymbol id="id71282" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id71286">π</m:ci>
															<m:times id="id71289"/>
														</m:apply>
														<m:ci id="id71290">a</m:ci>
													</m:apply>
													<m:apply id="id71292">
														<m:times id="id71293"/>
														<m:apply id="id71294">
															<m:csymbol id="id71295" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id71300">i</m:ci>
															<m:times id="id71302"/>
														</m:apply>
														<m:apply id="id71303">
															<m:ci id="id71304">~</m:ci>
															<m:ci id="id71306">a</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="theorem definition" id="S2.E1">
								<h6 id="id71318" class="title theorem-title bold">
									<span id="id71322" class="tag">Definition 2.1</span>.</h6>
								<div class="para" id="S2.E1.p1">
									<p id="id71335" style="" class="p">Given a weight <m:math id="id71341" display="inline">
											<m:semantics id="id71345">
												<m:mi id="id71346">α</m:mi>
												<m:annotation-xml id="id71348" encoding="MathML-Content">
													<m:ci id="id71351">α</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> of <m:math id="id71355" display="inline">
											<m:semantics id="id71358">
												<m:mi id="id71360">T</m:mi>
												<m:annotation-xml id="id71362" encoding="MathML-Content">
													<m:ci id="id71365">T</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, then we define <m:math id="id71368" display="inline">
											<m:semantics id="id71372">
												<m:mrow id="id71373">
													<m:mi id="id71374">e</m:mi>
													<m:mo id="id71376">⁢</m:mo>
													<m:mfenced id="id71378" open="(" close=")">
														<m:mi id="id71384">α</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id71386" encoding="MathML-Content">
													<m:apply id="id71389">
														<m:times id="id71390"/>
														<m:ci id="id71391">e</m:ci>
														<m:ci id="id71394">α</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> to be the Euler class</p>
									<table class="equation" id="S2.Ex4">

										<tr valign="baseline" class="equation" id="S2.Ex4">
											<td id="id71414" class="eqpad"/>
											<td id="id71418" nowrap="yes" align="center" colspan="1">
												<m:math id="id71426" display="block">
													<m:semantics id="id71429">
														<m:mrow id="id71430">
															<m:mi id="id71431">e</m:mi>
															<m:mo id="id71433">(</m:mo>
															<m:mi id="id71435">α</m:mi>
															<m:mo id="id71438">)</m:mo>
															<m:mo id="id71440">:=</m:mo>
															<m:mi id="id71442">e</m:mi>
															<m:mo id="id71444">(</m:mo>
															<m:msub id="id71446">
																<m:mi id="id71447">L</m:mi>
																<m:mi id="id71450">α</m:mi>
															</m:msub>
															<m:mo id="id71452">)</m:mo>
															<m:mo id="id71454">∈</m:mo>
															<m:msup id="id71456">
																<m:mi id="id71458">H</m:mi>
																<m:mo id="id71460">*</m:mo>
															</m:msup>
															<m:mo id="id71462">(</m:mo>
															<m:mi id="id71464">X</m:mi>
															<m:mo id="id71466">/</m:mo>
															<m:mo id="id71468">/</m:mo>
															<m:mi id="id71470">T</m:mi>
															<m:mo id="id71472">)</m:mo>
															<m:mo id="id71475">.</m:mo>
														</m:mrow>
														<m:annotation-xml id="id71477" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id71481" class="eqpad"/>
										</tr>

									</table>
									<p id="id71488" style="" class="p">(These are the cohomology classes which appear in Theorems A and B).</p>
								</div>

							</div>

							<div class="para" id="S2.SS3.p2">
								<p id="id71505" style="" class="p">We are now ready to prove theorem B (as stated in the introduction).
This integration formula is stated in terms of cohomology classes on
									<m:math id="id71514" display="inline">
										<m:semantics id="id71517">
											<m:mrow id="id71518">
												<m:mi id="id71519">X</m:mi>
												<m:mo id="id71521">/</m:mo>
												<m:mo id="id71523">/</m:mo>
												<m:mi id="id71525">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id71528" encoding="MathML-Content">
												<m:ci id="id71531">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id71534" display="inline">
										<m:semantics id="id71538">
											<m:mrow id="id71539">
												<m:mi id="id71540">X</m:mi>
												<m:mo id="id71542">/</m:mo>
												<m:mo id="id71544">/</m:mo>
												<m:mi id="id71546">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id71548" encoding="MathML-Content">
												<m:ci id="id71552">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. However, such classes often arise from equivariant
cohomology classes on <m:math id="id71556" display="inline">
										<m:semantics id="id71559">
											<m:mi id="id71560">X</m:mi>
											<m:annotation-xml id="id71562" encoding="MathML-Content">
												<m:ci id="id71566">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> via the ‘Kirwan map’. After proving the
integration formula, we will set up notation for equivariant
cohomology and state a corollary in terms of equivariant classes.</p>
							</div>

							<div id="id71574" class="proof">
								<h6 id="id71578" class="title proof-title italic">Proof of Theorem B.</h6>
								<div class="para" id="S2.SS3.p3">
									<p id="id71591" style="" class="p">Define the class <m:math id="id71597" display="inline">
											<m:semantics id="id71601">
												<m:mrow id="id71602">
													<m:mi id="id71603">b</m:mi>
													<m:mo id="id71605">∈</m:mo>
													<m:msup id="id71607">
														<m:mi id="id71608">H</m:mi>
														<m:mo id="id71611">*</m:mo>
													</m:msup>
													<m:mo id="id71613">(</m:mo>
													<m:mi id="id71615">X</m:mi>
													<m:mo id="id71617">/</m:mo>
													<m:mo id="id71619">/</m:mo>
													<m:mi id="id71621">T</m:mi>
													<m:mo id="id71623">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id71626" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> by <m:math id="id71630" display="inline">
											<m:semantics id="id71634">
												<m:mrow id="id71635">
													<m:mi id="id71636">b</m:mi>
													<m:mo id="id71638">:=</m:mo>
													<m:mrow id="id71640">
														<m:msub id="id71641">
															<m:mo id="id71642">∏</m:mo>
															<m:mrow id="id71644">
																<m:mi id="id71646">α</m:mi>
																<m:mo id="id71648">∈</m:mo>
																<m:msup id="id71650">
																	<m:mi id="id71651" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id71656">+</m:mo>
																</m:msup>
															</m:mrow>
														</m:msub>
														<m:mi id="id71658">e</m:mi>
														<m:mo id="id71660">⁢</m:mo>
														<m:mfenced id="id71663" open="(" close=")">
															<m:msub id="id71668">
																<m:mi id="id71669">L</m:mi>
																<m:mi id="id71671">α</m:mi>
															</m:msub>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id71673" encoding="MathML-Content">
													<m:apply id="id71677">
														<m:ci id="id71678">:=</m:ci>
														<m:ci id="id71680">b</m:ci>
														<m:apply id="id71682">
															<m:apply id="id71683">
																<m:csymbol id="id71684" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id71689">∏</m:ci>
																<m:apply id="id71691">
																	<m:ci id="id71692">∈</m:ci>
																	<m:ci id="id71695">α</m:ci>
																	<m:apply id="id71697">
																		<m:csymbol id="id71698" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id71703">Δ</m:ci>
																		<m:plus id="id71705"/>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply id="id71706">
																<m:times id="id71707"/>
																<m:ci id="id71708">e</m:ci>
																<m:apply id="id71711">
																	<m:csymbol id="id71712" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id71716">L</m:ci>
																	<m:ci id="id71718">α</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>. Then <m:math id="id71722" display="inline">
											<m:semantics id="id71726">
												<m:mrow id="id71727">
													<m:mrow id="id71728">
														<m:msup id="id71729">
															<m:mi id="id71730">i</m:mi>
															<m:mo id="id71732">*</m:mo>
														</m:msup>
														<m:mo id="id71734">⁢</m:mo>
														<m:mi id="id71736">b</m:mi>
													</m:mrow>
													<m:mo id="id71739">=</m:mo>
													<m:mrow id="id71741">
														<m:mi id="id71742">e</m:mi>
														<m:mo id="id71744">⁢</m:mo>
														<m:mfenced id="id71746" open="(" close=")">
															<m:mrow id="id71751">
																<m:mrow id="id71752">
																	<m:mi id="id71754">v</m:mi>
																	<m:mo id="id71756">⁢</m:mo>
																	<m:mi id="id71758">e</m:mi>
																	<m:mo id="id71760">⁢</m:mo>
																	<m:mi id="id71763">r</m:mi>
																	<m:mo id="id71765">⁢</m:mo>
																	<m:mi id="id71767">t</m:mi>
																</m:mrow>
																<m:mo id="id71769">⁢</m:mo>
																<m:mfenced id="id71772" open="(" close=")">
																	<m:mi id="id71777">π</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id71779" encoding="MathML-Content">
													<m:apply id="id71782">
														<m:eq id="id71784"/>
														<m:apply id="id71785">
															<m:times id="id71786"/>
															<m:apply id="id71787">
																<m:csymbol id="id71788" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id71792">i</m:ci>
																<m:times id="id71795"/>
															</m:apply>
															<m:ci id="id71796">b</m:ci>
														</m:apply>
														<m:apply id="id71798">
															<m:times id="id71799"/>
															<m:ci id="id71800">e</m:ci>
															<m:apply id="id71802">
																<m:times id="id71803"/>
																<m:ci id="id71804"/>
																<m:ci id="id71805">π</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, by proposition <a id="id71811" href="#S1.E2" title="Proposition 1.2 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
											<span id="id71809" style="" class="text tag">1.2</span>
										</a>.
We can calculate the integral over the fibres <m:math id="id71826" display="inline">
											<m:semantics id="id71829">
												<m:mrow id="id71830">
													<m:msub id="id71831">
														<m:mi id="id71832">π</m:mi>
														<m:mo id="id71835">*</m:mo>
													</m:msub>
													<m:mo id="id71837">(</m:mo>
													<m:msup id="id71839">
														<m:mi id="id71840">i</m:mi>
														<m:mo id="id71842">*</m:mo>
													</m:msup>
													<m:mi id="id71844">b</m:mi>
													<m:mo id="id71846">)</m:mo>
													<m:mo id="id71848">∈</m:mo>
													<m:msup id="id71851">
														<m:mi id="id71852">H</m:mi>
														<m:mn id="id71854">0</m:mn>
													</m:msup>
													<m:mo id="id71856">(</m:mo>
													<m:mi id="id71858">X</m:mi>
													<m:mo id="id71860">/</m:mo>
													<m:mo id="id71863">/</m:mo>
													<m:mi id="id71865">G</m:mi>
													<m:mo id="id71867">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id71869" encoding="MathML-Content"/>
											</m:semantics>
										</m:math> by restricting to a single fibre <m:math id="id71874" display="inline">
											<m:semantics id="id71877">
												<m:mrow id="id71878">
													<m:mi id="id71879">G</m:mi>
													<m:mo id="id71881">/</m:mo>
													<m:mi id="id71884">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id71886" encoding="MathML-Content">
													<m:apply id="id71889">
														<m:divide id="id71890"/>
														<m:ci id="id71891">G</m:ci>
														<m:ci id="id71893">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, using the
restriction property of the pushforward.  By naturality of the Euler
class, <m:math id="id71897" display="inline">
											<m:semantics id="id71901">
												<m:mrow id="id71902">
													<m:mrow id="id71903">
														<m:mi id="id71904">e</m:mi>
														<m:mo id="id71906">⁢</m:mo>
														<m:msub id="id71908">
															<m:mfenced id="id71910" open="(" close=")">
																<m:mrow id="id71915">
																	<m:mrow id="id71916">
																		<m:mi id="id71917">v</m:mi>
																		<m:mo id="id71919">⁢</m:mo>
																		<m:mi id="id71921">e</m:mi>
																		<m:mo id="id71923">⁢</m:mo>
																		<m:mi id="id71926">r</m:mi>
																		<m:mo id="id71928">⁢</m:mo>
																		<m:mi id="id71930">t</m:mi>
																	</m:mrow>
																	<m:mo id="id71932">⁢</m:mo>
																	<m:mfenced id="id71935" open="(" close=")">
																		<m:mi id="id71940">π</m:mi>
																	</m:mfenced>
																</m:mrow>
															</m:mfenced>
															<m:mrow id="id71942">
																<m:mi id="id71943">G</m:mi>
																<m:mo id="id71946">/</m:mo>
																<m:mi id="id71948">T</m:mi>
															</m:mrow>
														</m:msub>
													</m:mrow>
													<m:mo id="id71950">=</m:mo>
													<m:mrow id="id71952">
														<m:mi id="id71953">e</m:mi>
														<m:mo id="id71955">⁢</m:mo>
														<m:mfenced id="id71958" open="(" close=")">
															<m:mrow id="id71963">
																<m:mrow id="id71964">
																	<m:mi id="id71965">v</m:mi>
																	<m:mo id="id71967">⁢</m:mo>
																	<m:mi id="id71969">e</m:mi>
																	<m:mo id="id71971">⁢</m:mo>
																	<m:mi id="id71974">r</m:mi>
																	<m:mo id="id71976">⁢</m:mo>
																	<m:mi id="id71978">t</m:mi>
																</m:mrow>
																<m:mo id="id71980">⁢</m:mo>
																<m:msub id="id71983">
																	<m:mfenced id="id71984" open="(" close=")">
																		<m:mi id="id71989">π</m:mi>
																	</m:mfenced>
																	<m:mrow id="id71991">
																		<m:mi id="id71992">G</m:mi>
																		<m:mo id="id71995">/</m:mo>
																		<m:mi id="id71997">T</m:mi>
																	</m:mrow>
																</m:msub>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id71999">=</m:mo>
													<m:mrow id="id72001">
														<m:mi id="id72002">e</m:mi>
														<m:mo id="id72004">⁢</m:mo>
														<m:mfenced id="id72007" open="(" close=")">
															<m:mrow id="id72012">
																<m:mi id="id72013">T</m:mi>
																<m:mo id="id72015">⁢</m:mo>
																<m:mfenced id="id72017" open="(" close=")">
																	<m:mrow id="id72022">
																		<m:mi id="id72023">G</m:mi>
																		<m:mo id="id72026">/</m:mo>
																		<m:mi id="id72028">T</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id72030" encoding="MathML-Content">
													<m:apply id="id72033">
														<m:ci id="id72034"/>
														<m:apply id="id72035">
															<m:times id="id72036"/>
															<m:ci id="id72037">e</m:ci>
															<m:apply id="id72040">
																<m:csymbol id="id72041" cd="ambiguous">subscript</m:csymbol>
																<m:apply id="id72045">
																	<m:times id="id72046"/>
																	<m:ci id="id72047"/>
																	<m:ci id="id72048">π</m:ci>
																</m:apply>
																<m:apply id="id72051">
																	<m:divide id="id72052"/>
																	<m:ci id="id72053">G</m:ci>
																	<m:ci id="id72055">T</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq id="id72057"/>
														<m:apply id="id72058">
															<m:times id="id72059"/>
															<m:ci id="id72060">e</m:ci>
															<m:apply id="id72063">
																<m:times id="id72064"/>
																<m:ci id="id72065"/>
																<m:apply id="id72066">
																	<m:csymbol id="id72067" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id72072">π</m:ci>
																	<m:apply id="id72074">
																		<m:divide id="id72075"/>
																		<m:ci id="id72076">G</m:ci>
																		<m:ci id="id72078">T</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq id="id72080"/>
														<m:apply id="id72081">
															<m:times id="id72082"/>
															<m:ci id="id72084">e</m:ci>
															<m:apply id="id72086">
																<m:times id="id72087"/>
																<m:ci id="id72088">T</m:ci>
																<m:apply id="id72090">
																	<m:divide id="id72091"/>
																	<m:ci id="id72092">G</m:ci>
																	<m:ci id="id72094">T</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and so <m:math id="id72098" display="inline">
											<m:semantics id="id72101">
												<m:mrow id="id72102">
													<m:mrow id="id72103">
														<m:msub id="id72104">
															<m:mi id="id72105">π</m:mi>
															<m:mo id="id72108">*</m:mo>
														</m:msub>
														<m:mo id="id72110">⁢</m:mo>
														<m:mfenced id="id72112" open="(" close=")">
															<m:mrow id="id72117">
																<m:msup id="id72118">
																	<m:mi id="id72119">i</m:mi>
																	<m:mo id="id72122">*</m:mo>
																</m:msup>
																<m:mo id="id72124">⁢</m:mo>
																<m:mi id="id72126">b</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id72128">=</m:mo>
													<m:mrow id="id72130">
														<m:msub id="id72131">
															<m:mo id="id72132">∫</m:mo>
															<m:mrow id="id72135">
																<m:mi id="id72136">G</m:mi>
																<m:mo id="id72138">/</m:mo>
																<m:mi id="id72140">T</m:mi>
															</m:mrow>
														</m:msub>
														<m:mi id="id72142">e</m:mi>
														<m:mo id="id72144">⁢</m:mo>
														<m:mfenced id="id72147" open="(" close=")">
															<m:mrow id="id72152">
																<m:mi id="id72153">T</m:mi>
																<m:mo id="id72155">⁢</m:mo>
																<m:mfenced id="id72158" open="(" close=")">
																	<m:mrow id="id72163">
																		<m:mi id="id72164">G</m:mi>
																		<m:mo id="id72166">/</m:mo>
																		<m:mi id="id72168">T</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id72170">=</m:mo>
													<m:mrow id="id72172">
														<m:mi id="id72173">χ</m:mi>
														<m:mo id="id72176">⁢</m:mo>
														<m:mfenced id="id72178" open="(" close=")">
															<m:mrow id="id72183">
																<m:mi id="id72184">G</m:mi>
																<m:mo id="id72186">/</m:mo>
																<m:mi id="id72188">T</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id72191">=</m:mo>
													<m:mfenced id="id72193" open="|" close="|">
														<m:mi id="id72198">W</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id72200" encoding="MathML-Content">
													<m:apply id="id72203">
														<m:ci id="id72204"/>
														<m:apply id="id72205">
															<m:times id="id72206"/>
															<m:apply id="id72208">
																<m:csymbol id="id72209" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id72213">π</m:ci>
																<m:times id="id72216"/>
															</m:apply>
															<m:apply id="id72217">
																<m:times id="id72218"/>
																<m:apply id="id72219">
																	<m:csymbol id="id72220" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id72225">i</m:ci>
																	<m:times id="id72227"/>
																</m:apply>
																<m:ci id="id72228">b</m:ci>
															</m:apply>
														</m:apply>
														<m:eq id="id72230"/>
														<m:apply id="id72231">
															<m:apply id="id72232">
																<m:csymbol id="id72233" cd="ambiguous">subscript</m:csymbol>
																<m:int id="id72238"/>
																<m:apply id="id72239">
																	<m:divide id="id72240"/>
																	<m:ci id="id72241">G</m:ci>
																	<m:ci id="id72243">T</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id72245">
																<m:times id="id72246"/>
																<m:ci id="id72247">e</m:ci>
																<m:apply id="id72250">
																	<m:times id="id72251"/>
																	<m:ci id="id72252">T</m:ci>
																	<m:apply id="id72254">
																		<m:divide id="id72255"/>
																		<m:ci id="id72256">G</m:ci>
																		<m:ci id="id72258">T</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:eq id="id72260"/>
														<m:apply id="id72261">
															<m:times id="id72262"/>
															<m:ci id="id72263">χ</m:ci>
															<m:apply id="id72266">
																<m:divide id="id72267"/>
																<m:ci id="id72268">G</m:ci>
																<m:ci id="id72270">T</m:ci>
															</m:apply>
														</m:apply>
														<m:eq id="id72272"/>
														<m:apply id="id72273">
															<m:abs id="id72274"/>
															<m:ci id="id72275">W</m:ci>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, using a standard identification of the Euler characteristic of
										<m:math id="id72279" display="inline">
											<m:semantics id="id72283">
												<m:mrow id="id72284">
													<m:mi id="id72285">G</m:mi>
													<m:mo id="id72287">/</m:mo>
													<m:mi id="id72289">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id72291" encoding="MathML-Content">
													<m:apply id="id72294">
														<m:divide id="id72296"/>
														<m:ci id="id72297">G</m:ci>
														<m:ci id="id72299">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> <cite id="id72302" class="cite">[<a id="id72307" href="#bib.bib3" title="" class="ref">3</a>]</cite>.</p>
								</div>

								<div class="para" id="S2.SS3.p4">
									<p id="id72326" style="" class="p">Then</p>
									<table class="equation" id="S2.Ex5">

										<tr valign="baseline" class="equation" id="S2.Ex5">
											<td id="id72349" class="eqpad"/>
											<td id="id72353" nowrap="yes" align="center" colspan="1">
												<m:math id="id72361" display="block">
													<m:semantics id="id72364">
														<m:mrow id="id72365">
															<m:msub id="id72366">
																<m:mo id="id72367">∫</m:mo>
																<m:mrow id="id72370">
																	<m:mi id="id72371">X</m:mi>
																	<m:mo id="id72373">⁣</m:mo>
																	<m:mo id="id72375">/</m:mo>
																	<m:mo id="id72377">⁣</m:mo>
																	<m:mrow id="id72380">
																		<m:none id="id72381"/>
																		<m:mo id="id72382">/</m:mo>
																		<m:mi id="id72384">G</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mi id="id72386">a</m:mi>
															<m:mo id="id72388">=</m:mo>
															<m:mfrac id="id72390">
																<m:mn id="id72392">1</m:mn>
																<m:mfenced id="id72394" open="|" close="|">
																	<m:mi id="id72399">W</m:mi>
																</m:mfenced>
															</m:mfrac>
															<m:msub id="id72401">
																<m:mo id="id72402">∫</m:mo>
																<m:mrow id="id72404">
																	<m:mrow id="id72405">
																		<m:msubsup id="id72406">
																			<m:mi id="id72408">μ</m:mi>
																			<m:mi id="id72410">G</m:mi>
																			<m:mrow id="id72412">
																				<m:mo id="id72413">-</m:mo>
																				<m:mn id="id72415">1</m:mn>
																			</m:mrow>
																		</m:msubsup>
																		<m:mo id="id72417">⁢</m:mo>
																		<m:mfenced id="id72420" open="(" close=")">
																			<m:mn id="id72425">0</m:mn>
																		</m:mfenced>
																	</m:mrow>
																	<m:mo id="id72427">/</m:mo>
																	<m:mi id="id72429">T</m:mi>
																</m:mrow>
															</m:msub>
															<m:msup id="id72431">
																<m:mi id="id72432">π</m:mi>
																<m:mo id="id72435">*</m:mo>
															</m:msup>
															<m:mi id="id72437">a</m:mi>
															<m:mo id="id72439">⌣</m:mo>
															<m:msup id="id72441">
																<m:mi id="id72442">i</m:mi>
																<m:mo id="id72445">*</m:mo>
															</m:msup>
															<m:mi id="id72447">b</m:mi>
															<m:mo id="id72449">,</m:mo>
															<m:mtext id="id72451">applying the push-pull formula,</m:mtext>
															<m:mo id="id72454">=</m:mo>
															<m:mfrac id="id72456">
																<m:mn id="id72457">1</m:mn>
																<m:mfenced id="id72459" open="|" close="|">
																	<m:mi id="id72464">W</m:mi>
																</m:mfenced>
															</m:mfrac>
															<m:msub id="id72466">
																<m:mo id="id72467">∫</m:mo>
																<m:mrow id="id72470">
																	<m:mi id="id72471">X</m:mi>
																	<m:mo id="id72473">⁣</m:mo>
																	<m:mo id="id72475">/</m:mo>
																	<m:mo id="id72477">⁣</m:mo>
																	<m:mrow id="id72480">
																		<m:none id="id72481"/>
																		<m:mo id="id72482">/</m:mo>
																		<m:mi id="id72484">T</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:msub id="id72486">
																<m:mi id="id72487">i</m:mi>
																<m:mo id="id72489">*</m:mo>
															</m:msub>
															<m:msup id="id72491">
																<m:mi id="id72492">i</m:mi>
																<m:mo id="id72495">*</m:mo>
															</m:msup>
															<m:mo id="id72497">(</m:mo>
															<m:mover id="id72499" accent="true">
																<m:mi id="id72502">a</m:mi>
																<m:mo id="id72504">~</m:mo>
															</m:mover>
															<m:mo id="id72506">⌣</m:mo>
															<m:mi id="id72509">b</m:mi>
															<m:mo id="id72511">)</m:mo>
															<m:mo id="id72513">,</m:mo>
															<m:mrow id="id72515">
																<m:mtext id="id72516">composition of pushforwards; and </m:mtext>
																<m:mover id="id72519" accent="true">
																	<m:mi id="id72522">a</m:mi>
																	<m:mo id="id72525">~</m:mo>
																</m:mover>
																<m:mtext id="id72527"> is a lift of </m:mtext>
																<m:mi id="id72529">a</m:mi>
																<m:mtext id="id72531">,</m:mtext>
															</m:mrow>
															<m:mo id="id72533">=</m:mo>
															<m:mfrac id="id72536">
																<m:mn id="id72537">1</m:mn>
																<m:mfenced id="id72539" open="|" close="|">
																	<m:mi id="id72544">W</m:mi>
																</m:mfenced>
															</m:mfrac>
															<m:msub id="id72546">
																<m:mo id="id72547">∫</m:mo>
																<m:mrow id="id72549">
																	<m:mi id="id72550">X</m:mi>
																	<m:mo id="id72553">⁣</m:mo>
																	<m:mo id="id72555">/</m:mo>
																	<m:mo id="id72557">⁣</m:mo>
																	<m:mrow id="id72560">
																		<m:none id="id72561"/>
																		<m:mo id="id72562">/</m:mo>
																		<m:mi id="id72564">T</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mover id="id72566" accent="true">
																<m:mi id="id72569">a</m:mi>
																<m:mo id="id72571">~</m:mo>
															</m:mover>
															<m:mo id="id72574">⌣</m:mo>
															<m:mi id="id72576">b</m:mi>
															<m:mo id="id72578">⌣</m:mo>
															<m:munder id="id72580">
																<m:mo id="id72582" movablelimits="false">∏</m:mo>
																<m:mrow id="id72586">
																	<m:mi id="id72587">α</m:mi>
																	<m:mo id="id72590">∈</m:mo>
																	<m:msup id="id72592">
																		<m:mi id="id72593" mathvariant="normal">Δ</m:mi>
																		<m:mo id="id72598">-</m:mo>
																	</m:msup>
																</m:mrow>
															</m:munder>
															<m:mi id="id72600">e</m:mi>
															<m:mo id="id72602">(</m:mo>
															<m:msub id="id72604">
																<m:mi id="id72605">L</m:mi>
																<m:mi id="id72607">α</m:mi>
															</m:msub>
															<m:mo id="id72610">)</m:mo>
															<m:mo id="id72612">,</m:mo>
															<m:mrow id="id72614">
																<m:mtext id="id72615">properties of the Euler class, applied to the</m:mtext>
																<m:mtext id="id72618"> vector bundle identified in proposition </m:mtext>
																<m:mtext id="id72620">1.2</m:mtext>
																<m:mtext id="id72623">(1),</m:mtext>
															</m:mrow>
															<m:mo id="id72625">=</m:mo>
															<m:mfrac id="id72627">
																<m:mn id="id72628">1</m:mn>
																<m:mfenced id="id72630" open="|" close="|">
																	<m:mi id="id72635">W</m:mi>
																</m:mfenced>
															</m:mfrac>
															<m:msub id="id72638">
																<m:mo id="id72639">∫</m:mo>
																<m:mrow id="id72641">
																	<m:mi id="id72642">X</m:mi>
																	<m:mo id="id72644">⁣</m:mo>
																	<m:mo id="id72647">/</m:mo>
																	<m:mo id="id72649">⁣</m:mo>
																	<m:mrow id="id72651">
																		<m:none id="id72652"/>
																		<m:mo id="id72653">/</m:mo>
																		<m:mi id="id72655">T</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mover id="id72658" accent="true">
																<m:mi id="id72661">a</m:mi>
																<m:mo id="id72663">~</m:mo>
															</m:mover>
															<m:mo id="id72665">⌣</m:mo>
															<m:mi id="id72668">e</m:mi>
															<m:mo id="id72670">,</m:mo>
															<m:mrow id="id72672">
																<m:mtext id="id72673">by definition of </m:mtext>
																<m:mi id="id72675">e</m:mi>
																<m:mtext id="id72678">.</m:mtext>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id72680" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id72684" class="eqpad"/>
										</tr>

									</table>
									<p id="id72691" style="" class="p">∎</p>
								</div>

							</div>

						</div>

					</div>

					<div class="section" id="S3">
						<h2 id="id72709" class="title section-title">
							<span id="id72712" class="tag">3 </span>The relationship between the cohomology rings</h2>
						<div class="para" id="S3.p1">
							<p id="id72726" style="" class="p">In this section we prove theorem A, which relates the cohomology rings
of the  symplectic quotients <m:math id="id72734" display="inline">
									<m:semantics id="id72737">
										<m:mrow id="id72738">
											<m:mi id="id72739">X</m:mi>
											<m:mo id="id72741">/</m:mo>
											<m:mo id="id72743">/</m:mo>
											<m:mi id="id72745">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id72748" encoding="MathML-Content">
											<m:ci id="id72751">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id72754" display="inline">
									<m:semantics id="id72758">
										<m:mrow id="id72759">
											<m:mi id="id72760">X</m:mi>
											<m:mo id="id72762">/</m:mo>
											<m:mo id="id72764">/</m:mo>
											<m:mi id="id72766">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id72768" encoding="MathML-Content">
											<m:ci id="id72772">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. The proof involves
some standard machinery in equivariant cohomology, and a crucial
result concerning the ‘Kirwan map’, which we begin by reviewing.
We also state a version of the integration formula in the language of
equivariant cohomology.</p>
						</div>

						<div class="subsection" id="S3.SS1">
							<h3 id="id72788" class="title subsection-title">
								<span id="id72791" class="tag">3.1 </span>Some key facts in equivariant cohomology</h3>
							<div class="para" id="S3.SS1.p1">
								<p id="id72805" style="" class="p">The <m:math id="id72811" display="inline">
										<m:semantics id="id72815">
											<m:mi id="id72816">G</m:mi>
											<m:annotation-xml id="id72818" encoding="MathML-Content">
												<m:ci id="id72821">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-equivariant cohomology of the <m:math id="id72825" display="inline">
										<m:semantics id="id72828">
											<m:mi id="id72829">G</m:mi>
											<m:annotation-xml id="id72831" encoding="MathML-Content">
												<m:ci id="id72835">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-manifold <m:math id="id72838" display="inline">
										<m:semantics id="id72842">
											<m:mi id="id72843">X</m:mi>
											<m:annotation-xml id="id72845" encoding="MathML-Content">
												<m:ci id="id72848">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which we
denote by <m:math id="id72852" display="inline">
										<m:semantics id="id72855">
											<m:mrow id="id72856">
												<m:msubsup id="id72857">
													<m:mi id="id72858">H</m:mi>
													<m:mi id="id72860">G</m:mi>
													<m:mo id="id72862">*</m:mo>
												</m:msubsup>
												<m:mo id="id72864">⁢</m:mo>
												<m:mfenced id="id72867" open="(" close=")">
													<m:mi id="id72872">X</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id72874" encoding="MathML-Content">
												<m:apply id="id72877">
													<m:times id="id72878"/>
													<m:apply id="id72880">
														<m:csymbol id="id72881" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id72885">
															<m:csymbol id="id72886" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id72891">H</m:ci>
															<m:ci id="id72893">G</m:ci>
														</m:apply>
														<m:times id="id72895"/>
													</m:apply>
													<m:ci id="id72896">X</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, is defined to be the ordinary cohomology of the
									<span id="id72900" style="" class="text bold">homotopy quotient</span>
								</p>
								<table class="equation" id="S3.Ex1">

									<tr valign="baseline" class="equation" id="S3.Ex1">
										<td id="id72924" class="eqpad"/>
										<td id="id72929" nowrap="yes" align="center" colspan="1">
											<m:math id="id72936" display="block">
												<m:semantics id="id72940">
													<m:mrow id="id72941">
														<m:mrow id="id72942">
															<m:msub id="id72943">
																<m:mi id="id72944">X</m:mi>
																<m:mi id="id72946">G</m:mi>
															</m:msub>
															<m:mo id="id72948">:=</m:mo>
															<m:mrow id="id72950">
																<m:mfenced id="id72951" open="(" close=")">
																	<m:mrow id="id72956">
																		<m:mrow id="id72957">
																			<m:mi id="id72958">E</m:mi>
																			<m:mo id="id72961">⁢</m:mo>
																			<m:mi id="id72963">G</m:mi>
																		</m:mrow>
																		<m:mo id="id72965">×</m:mo>
																		<m:mi id="id72968">X</m:mi>
																	</m:mrow>
																</m:mfenced>
																<m:mo id="id72970">/</m:mo>
																<m:mi id="id72972">G</m:mi>
															</m:mrow>
														</m:mrow>
														<m:mo id="id72974">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id72976" encoding="MathML-Content">
														<m:apply id="id72979">
															<m:ci id="id72980">:=</m:ci>
															<m:apply id="id72983">
																<m:csymbol id="id72984" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id72988">X</m:ci>
																<m:ci id="id72990">G</m:ci>
															</m:apply>
															<m:apply id="id72993">
																<m:divide id="id72994"/>
																<m:apply id="id72995">
																	<m:times id="id72996"/>
																	<m:apply id="id72997">
																		<m:times id="id72998"/>
																		<m:ci id="id72999">E</m:ci>
																		<m:ci id="id73001">G</m:ci>
																	</m:apply>
																	<m:ci id="id73003">X</m:ci>
																</m:apply>
																<m:ci id="id73005">G</m:ci>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id73009" class="eqpad"/>
									</tr>

								</table>
								<p id="id73015" style="" class="p">where <m:math id="id73022" display="inline">
										<m:semantics id="id73025">
											<m:mrow id="id73026">
												<m:mi id="id73027">E</m:mi>
												<m:mo id="id73029">⁢</m:mo>
												<m:mi id="id73032">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id73034" encoding="MathML-Content">
												<m:apply id="id73037">
													<m:times id="id73038"/>
													<m:ci id="id73039">E</m:ci>
													<m:ci id="id73041">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a universal space for <m:math id="id73045" display="inline">
										<m:semantics id="id73048">
											<m:mi id="id73049">G</m:mi>
											<m:annotation-xml id="id73051" encoding="MathML-Content">
												<m:ci id="id73055">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>: that is, <m:math id="id73058" display="inline">
										<m:semantics id="id73062">
											<m:mrow id="id73063">
												<m:mi id="id73064">E</m:mi>
												<m:mo id="id73066">⁢</m:mo>
												<m:mi id="id73068">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id73070" encoding="MathML-Content">
												<m:apply id="id73074">
													<m:times id="id73075"/>
													<m:ci id="id73076">E</m:ci>
													<m:ci id="id73078">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is contractible
and has a free <m:math id="id73082" display="inline">
										<m:semantics id="id73085">
											<m:mi id="id73086">G</m:mi>
											<m:annotation-xml id="id73088" encoding="MathML-Content">
												<m:ci id="id73091">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action. For various facts in equivariant
cohomology,
see <cite id="id73096" class="cite">[<a id="id73100" href="#bib.bib1" title="" class="ref">1</a>, <a id="id73110" href="#bib.bib6" title="" class="ref">6</a>, <a id="id73119" href="#bib.bib16" title="" class="ref">16</a>]</cite>.
We recall that, if <m:math id="id73130" display="inline">
										<m:semantics id="id73134">
											<m:mrow id="id73135">
												<m:mi id="id73136">K</m:mi>
												<m:mo id="id73138">⊂</m:mo>
												<m:mi id="id73141">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id73143" encoding="MathML-Content">
												<m:apply id="id73146">
													<m:ci id="id73147">⊂</m:ci>
													<m:ci id="id73150">K</m:ci>
													<m:ci id="id73152">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a subgroup, there is a natural
restriction map <m:math id="id73155" display="inline">
										<m:semantics id="id73159">
											<m:mrow id="id73160">
												<m:msubsup id="id73161">
													<m:mi id="id73162">r</m:mi>
													<m:mi id="id73164">K</m:mi>
													<m:mi id="id73166">G</m:mi>
												</m:msubsup>
												<m:mo id="id73168">:</m:mo>
												<m:mrow id="id73170">
													<m:mrow id="id73172">
														<m:msubsup id="id73173">
															<m:mi id="id73174">H</m:mi>
															<m:mi id="id73176">G</m:mi>
															<m:mo id="id73178">*</m:mo>
														</m:msubsup>
														<m:mo id="id73180">⁢</m:mo>
														<m:mfenced id="id73182" open="(" close=")">
															<m:mi id="id73188">X</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id73190">→</m:mo>
													<m:mrow id="id73192">
														<m:msubsup id="id73193">
															<m:mi id="id73194">H</m:mi>
															<m:mi id="id73196">K</m:mi>
															<m:mo id="id73198">*</m:mo>
														</m:msubsup>
														<m:mo id="id73201">⁢</m:mo>
														<m:mfenced id="id73203" open="(" close=")">
															<m:mi id="id73208">X</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id73210" encoding="MathML-Content">
												<m:apply id="id73214">
													<m:ci id="id73215">:</m:ci>
													<m:apply id="id73217">
														<m:csymbol id="id73218" cd="ambiguous">subscript</m:csymbol>
														<m:apply id="id73222">
															<m:csymbol id="id73224" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id73228">r</m:ci>
															<m:ci id="id73230">G</m:ci>
														</m:apply>
														<m:ci id="id73232">K</m:ci>
													</m:apply>
													<m:apply id="id73235">
														<m:ci id="id73236">→</m:ci>
														<m:apply id="id73238">
															<m:times id="id73239"/>
															<m:apply id="id73240">
																<m:csymbol id="id73241" cd="ambiguous">superscript</m:csymbol>
																<m:apply id="id73246">
																	<m:csymbol id="id73247" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id73252">H</m:ci>
																	<m:ci id="id73254">G</m:ci>
																</m:apply>
																<m:times id="id73256"/>
															</m:apply>
															<m:ci id="id73257">X</m:ci>
														</m:apply>
														<m:apply id="id73259">
															<m:times id="id73260"/>
															<m:apply id="id73261">
																<m:csymbol id="id73262" cd="ambiguous">subscript</m:csymbol>
																<m:apply id="id73267">
																	<m:csymbol id="id73268" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id73273">H</m:ci>
																	<m:times id="id73275"/>
																</m:apply>
																<m:ci id="id73276">K</m:ci>
															</m:apply>
															<m:ci id="id73278">X</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S3.SS1.p2">
								<p id="id73290" style="" class="p">A map of fundamental importance in symplectic geometry is the
									<span id="id73297" style="" class="text bold">Kirwan map</span>, which gives a surjective ring homomorphism from the
equivariant cohomology of a symplectic manifold onto the ordinary
cohomology of its symplectic quotient.  Explicitly, we define the
Kirwan map</p>
								<table class="equation" id="S3.Ex2">

									<tr valign="baseline" class="equation" id="S3.Ex2">
										<td id="id73323" class="eqpad"/>
										<td id="id73327" nowrap="yes" align="center" colspan="1">
											<m:math id="id73335" display="block">
												<m:semantics id="id73338">
													<m:mrow id="id73339">
														<m:msub id="id73340">
															<m:mi id="id73341">κ</m:mi>
															<m:mi id="id73344">G</m:mi>
														</m:msub>
														<m:mo id="id73346">:</m:mo>
														<m:msubsup id="id73348">
															<m:mi id="id73349">H</m:mi>
															<m:mi id="id73351">G</m:mi>
															<m:mo id="id73353">*</m:mo>
														</m:msubsup>
														<m:mo id="id73355">(</m:mo>
														<m:mi id="id73358">X</m:mi>
														<m:mo id="id73360">)</m:mo>
														<m:mo id="id73362">→</m:mo>
														<m:msup id="id73364">
															<m:mi id="id73365">H</m:mi>
															<m:mo id="id73367">*</m:mo>
														</m:msup>
														<m:mo id="id73370">(</m:mo>
														<m:mi id="id73372">X</m:mi>
														<m:mo id="id73374">/</m:mo>
														<m:mo id="id73376">/</m:mo>
														<m:mi id="id73378">G</m:mi>
														<m:mo id="id73380">)</m:mo>
														<m:mo id="id73382">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id73384" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id73389" class="eqpad"/>
									</tr>

								</table>
								<p id="id73395" style="" class="p">by taking the restriction to <m:math id="id73402" display="inline">
										<m:semantics id="id73405">
											<m:mrow id="id73406">
												<m:msubsup id="id73407">
													<m:mi id="id73408">μ</m:mi>
													<m:mi id="id73411">G</m:mi>
													<m:mrow id="id73413">
														<m:mo id="id73414">-</m:mo>
														<m:mn id="id73416">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id73418">⁢</m:mo>
												<m:mfenced id="id73421" open="(" close=")">
													<m:mn id="id73426">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id73428" encoding="MathML-Content">
												<m:apply id="id73431">
													<m:times id="id73432"/>
													<m:apply id="id73433">
														<m:csymbol id="id73434" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id73439">
															<m:csymbol id="id73440" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id73445">μ</m:ci>
															<m:ci id="id73447">G</m:ci>
														</m:apply>
														<m:apply id="id73449">
															<m:minus id="id73450"/>
															<m:cn id="id73452">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id73454">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and composing this with the
natural isomorphism <span id="id73457" style="" class="text">
										<m:math id="id73463" display="inline">
											<m:semantics id="id73466">
												<m:mrow id="id73467">
													<m:msubsup id="id73468">
														<m:mi id="id73469">H</m:mi>
														<m:mi id="id73471">G</m:mi>
														<m:mo id="id73474">*</m:mo>
													</m:msubsup>
													<m:mo id="id73476">(</m:mo>
													<m:msubsup id="id73478">
														<m:mi id="id73479">μ</m:mi>
														<m:mi id="id73481">G</m:mi>
														<m:mrow id="id73483">
															<m:mo id="id73484">-</m:mo>
															<m:mn id="id73487">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id73489">(</m:mo>
													<m:mn id="id73491">0</m:mn>
													<m:mo id="id73493">)</m:mo>
													<m:mo id="id73495">)</m:mo>
													<m:mstyle id="id73497" scriptlevel="+1">
														<m:mtable id="id73500" rowspacing="0.2ex" columnspacing="0.4em">
															<m:mtr id="id73506">
																<m:mtd id="id73507">
																	<m:mo id="id73508">≅</m:mo>
																</m:mtd>
															</m:mtr>
															<m:mtr id="id73510">
																<m:mtd id="id73512">
																	<m:mi id="id73513">→</m:mi>
																</m:mtd>
															</m:mtr>
														</m:mtable>
													</m:mstyle>
													<m:msup id="id73515">
														<m:mi id="id73516">H</m:mi>
														<m:mo id="id73518">*</m:mo>
													</m:msup>
													<m:mo id="id73520">(</m:mo>
													<m:mi id="id73522">X</m:mi>
													<m:mo id="id73525">/</m:mo>
													<m:mo id="id73527">/</m:mo>
													<m:mi id="id73529">G</m:mi>
													<m:mo id="id73531">)</m:mo>
												</m:mrow>
												<m:annotation-xml id="id73533" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</span>. This
natural isomorphism is defined in rational cohomology whenever the
									<m:math id="id73538" display="inline">
										<m:semantics id="id73542">
											<m:mi id="id73543">G</m:mi>
											<m:annotation-xml id="id73545" encoding="MathML-Content">
												<m:ci id="id73548">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action on <m:math id="id73552" display="inline">
										<m:semantics id="id73555">
											<m:mrow id="id73556">
												<m:msubsup id="id73557">
													<m:mi id="id73558">μ</m:mi>
													<m:mi id="id73561">G</m:mi>
													<m:mrow id="id73563">
														<m:mo id="id73564">-</m:mo>
														<m:mn id="id73566">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id73568">⁢</m:mo>
												<m:mfenced id="id73570" open="(" close=")">
													<m:mn id="id73576">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id73578" encoding="MathML-Content">
												<m:apply id="id73581">
													<m:times id="id73582"/>
													<m:apply id="id73583">
														<m:csymbol id="id73584" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id73589">
															<m:csymbol id="id73590" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id73595">μ</m:ci>
															<m:ci id="id73597">G</m:ci>
														</m:apply>
														<m:apply id="id73599">
															<m:minus id="id73600"/>
															<m:cn id="id73601">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id73603">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is locally free, and we understand
									<m:math id="id73607" display="inline">
										<m:semantics id="id73610">
											<m:msub id="id73611">
												<m:mi id="id73612">κ</m:mi>
												<m:mi id="id73615">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id73617" encoding="MathML-Content">
												<m:apply id="id73620">
													<m:csymbol id="id73621" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id73626">κ</m:ci>
													<m:ci id="id73628">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to only be defined when this is the case.  We denote the
analogous map for the maximal torus <m:math id="id73633" display="inline">
										<m:semantics id="id73636">
											<m:mi id="id73637">T</m:mi>
											<m:annotation-xml id="id73639" encoding="MathML-Content">
												<m:ci id="id73642">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> by <m:math id="id73646" display="inline">
										<m:semantics id="id73649">
											<m:mrow id="id73650">
												<m:msub id="id73651">
													<m:mi id="id73652">κ</m:mi>
													<m:mi id="id73655">T</m:mi>
												</m:msub>
												<m:mo id="id73657">:</m:mo>
												<m:msubsup id="id73659">
													<m:mi id="id73660">H</m:mi>
													<m:mi id="id73662">T</m:mi>
													<m:mo id="id73664">*</m:mo>
												</m:msubsup>
												<m:mo id="id73667">(</m:mo>
												<m:mi id="id73669">X</m:mi>
												<m:mo id="id73671">)</m:mo>
												<m:mo id="id73673">→</m:mo>
												<m:msup id="id73675">
													<m:mi id="id73676">H</m:mi>
													<m:mo id="id73679">*</m:mo>
												</m:msup>
												<m:mo id="id73681">(</m:mo>
												<m:mi id="id73683">X</m:mi>
												<m:mo id="id73685">/</m:mo>
												<m:mo id="id73687">/</m:mo>
												<m:mi id="id73689">T</m:mi>
												<m:mo id="id73691">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id73694" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>.
								</p>
							</div>

							<div class="para" id="S3.SS1.p3">
								<p id="id73707" style="" class="p">We observe that, for any equivariant class <m:math id="id73713" display="inline">
										<m:semantics id="id73717">
											<m:mrow id="id73718">
												<m:mi id="id73719">a</m:mi>
												<m:mo id="id73721">∈</m:mo>
												<m:mrow id="id73723">
													<m:msubsup id="id73724">
														<m:mi id="id73726">H</m:mi>
														<m:mi id="id73728">G</m:mi>
														<m:mo id="id73730">*</m:mo>
													</m:msubsup>
													<m:mo id="id73732">⁢</m:mo>
													<m:mfenced id="id73734" open="(" close=")">
														<m:mi id="id73739">X</m:mi>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id73742" encoding="MathML-Content">
												<m:apply id="id73745">
													<m:ci id="id73746">∈</m:ci>
													<m:ci id="id73748">a</m:ci>
													<m:apply id="id73750">
														<m:times id="id73752"/>
														<m:apply id="id73753">
															<m:csymbol id="id73754" cd="ambiguous">subscript</m:csymbol>
															<m:apply id="id73758">
																<m:csymbol id="id73759" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id73764">H</m:ci>
																<m:times id="id73766"/>
															</m:apply>
															<m:ci id="id73767">G</m:ci>
														</m:apply>
														<m:ci id="id73769">X</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, the class
									<m:math id="id73773" display="inline">
										<m:semantics id="id73776">
											<m:mrow id="id73777">
												<m:mrow id="id73778">
													<m:msub id="id73779">
														<m:mi id="id73780">κ</m:mi>
														<m:mi id="id73783">T</m:mi>
													</m:msub>
													<m:mo id="id73785">∘</m:mo>
													<m:msubsup id="id73787">
														<m:mi id="id73788">r</m:mi>
														<m:mi id="id73791">T</m:mi>
														<m:mi id="id73793">G</m:mi>
													</m:msubsup>
												</m:mrow>
												<m:mo id="id73795">⁢</m:mo>
												<m:mfenced id="id73797" open="(" close=")">
													<m:mi id="id73802">a</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id73804" encoding="MathML-Content">
												<m:apply id="id73808">
													<m:times id="id73809"/>
													<m:apply id="id73810">
														<m:compose id="id73811"/>
														<m:apply id="id73812">
															<m:csymbol id="id73813" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id73818">κ</m:ci>
															<m:ci id="id73820">T</m:ci>
														</m:apply>
														<m:apply id="id73822">
															<m:csymbol id="id73823" cd="ambiguous">subscript</m:csymbol>
															<m:apply id="id73828">
																<m:csymbol id="id73829" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id73834">r</m:ci>
																<m:ci id="id73836">G</m:ci>
															</m:apply>
															<m:ci id="id73838">T</m:ci>
														</m:apply>
													</m:apply>
													<m:ci id="id73840">a</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a lift of <m:math id="id73844" display="inline">
										<m:semantics id="id73847">
											<m:mrow id="id73848">
												<m:msub id="id73849">
													<m:mi id="id73850">κ</m:mi>
													<m:mi id="id73853">G</m:mi>
												</m:msub>
												<m:mo id="id73855">⁢</m:mo>
												<m:mfenced id="id73857" open="(" close=")">
													<m:mi id="id73862">a</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id73864" encoding="MathML-Content">
												<m:apply id="id73868">
													<m:times id="id73869"/>
													<m:apply id="id73870">
														<m:csymbol id="id73871" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id73876">κ</m:ci>
														<m:ci id="id73878">G</m:ci>
													</m:apply>
													<m:ci id="id73880">a</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (see comments in the
proof of theorem A for more elucidation on this point). We can thus
restate the integration formula, theorem B, in a form which is more
natural in many applications:</p>
							</div>

							<div class="theorem corollary" id="S3.E1">
								<h6 id="id73895" class="title theorem-title bold">
									<span id="id73899" class="tag">Corollary 3.1</span> (Integration formula in terms of equivariant
cohomology).</h6>
								<div class="para" id="S3.E1.p1">
									<p id="id73913" style="" class="p">
										<span id="id73918" style="" class="text italic">For all <m:math id="id73924" display="inline">
												<m:semantics id="id73928">
													<m:mrow id="id73929">
														<m:mi id="id73930">a</m:mi>
														<m:mo id="id73932" mathvariant="normal">∈</m:mo>
														<m:mrow id="id73937">
															<m:msubsup id="id73938">
																<m:mi id="id73939">H</m:mi>
																<m:mi id="id73941">G</m:mi>
																<m:mo id="id73943" mathvariant="normal">*</m:mo>
															</m:msubsup>
															<m:mo id="id73948" mathvariant="italic">⁢</m:mo>
															<m:mfenced id="id73952" open="(" close=")">
																<m:mi id="id73957">X</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id73959" encoding="MathML-Content">
														<m:apply id="id73963">
															<m:ci id="id73964">∈</m:ci>
															<m:ci id="id73966">a</m:ci>
															<m:apply id="id73968">
																<m:times id="id73969"/>
																<m:apply id="id73970">
																	<m:csymbol id="id73972" cd="ambiguous">subscript</m:csymbol>
																	<m:apply id="id73976">
																		<m:csymbol id="id73977" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id73982">H</m:ci>
																		<m:times id="id73984"/>
																	</m:apply>
																	<m:ci id="id73985">G</m:ci>
																</m:apply>
																<m:ci id="id73987">X</m:ci>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>,</span>
									</p>
									<table class="equation" id="S3.Ex3">

										<tr valign="baseline" class="equation" id="S3.Ex3">
											<td id="id74007" class="eqpad"/>
											<td id="id74012" nowrap="yes" align="center" colspan="1">
												<m:math id="id74019" display="block">
													<m:semantics id="id74022">
														<m:mrow id="id74023">
															<m:mrow id="id74024">
																<m:mrow id="id74025">
																	<m:msub id="id74026">
																		<m:mo id="id74028">∫</m:mo>
																		<m:mrow id="id74030">
																			<m:mi id="id74031">X</m:mi>
																			<m:mo id="id74033">⁣</m:mo>
																			<m:mo id="id74036">/</m:mo>
																			<m:mo id="id74038">⁣</m:mo>
																			<m:mrow id="id74040">
																				<m:none id="id74041"/>
																				<m:mo id="id74042">/</m:mo>
																				<m:mi id="id74044">G</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msub>
																	<m:msub id="id74046">
																		<m:mi id="id74048">κ</m:mi>
																		<m:mi id="id74050">G</m:mi>
																	</m:msub>
																	<m:mo id="id74052">⁢</m:mo>
																	<m:mfenced id="id74054" open="(" close=")">
																		<m:mi id="id74060">a</m:mi>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id74062">=</m:mo>
																<m:mrow id="id74064">
																	<m:mfrac id="id74065">
																		<m:mn id="id74066">1</m:mn>
																		<m:mfenced id="id74068" open="|" close="|">
																			<m:mi id="id74073">W</m:mi>
																		</m:mfenced>
																	</m:mfrac>
																	<m:mo id="id74075">⁢</m:mo>
																	<m:mrow id="id74078">
																		<m:msub id="id74079">
																			<m:mo id="id74080">∫</m:mo>
																			<m:mrow id="id74082">
																				<m:mi id="id74083">X</m:mi>
																				<m:mo id="id74085">⁣</m:mo>
																				<m:mo id="id74088">/</m:mo>
																				<m:mo id="id74090">⁣</m:mo>
																				<m:mrow id="id74092">
																					<m:none id="id74093"/>
																					<m:mo id="id74094">/</m:mo>
																					<m:mi id="id74097">T</m:mi>
																				</m:mrow>
																			</m:mrow>
																		</m:msub>
																		<m:mrow id="id74099">
																			<m:msub id="id74100">
																				<m:mi id="id74101">κ</m:mi>
																				<m:mi id="id74103">T</m:mi>
																			</m:msub>
																			<m:mo id="id74105">∘</m:mo>
																			<m:msubsup id="id74108">
																				<m:mi id="id74109">r</m:mi>
																				<m:mi id="id74111">T</m:mi>
																				<m:mi id="id74113">G</m:mi>
																			</m:msubsup>
																		</m:mrow>
																		<m:mo id="id74115">⁢</m:mo>
																		<m:mfenced id="id74118" open="(" close=")">
																			<m:mi id="id74123">a</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:mrow>
																<m:mo id="id74125">⌣</m:mo>
																<m:mi id="id74127">e</m:mi>
															</m:mrow>
															<m:mo id="id74129">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id74132" encoding="MathML-Content">
															<m:apply id="id74135">
																<m:ci id="id74136"/>
																<m:apply id="id74137">
																	<m:apply id="id74138">
																		<m:csymbol id="id74139" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id74144"/>
																		<m:apply id="id74145">
																			<m:list id="id74146"/>
																			<m:ci id="id74147">X</m:ci>
																			<m:divide id="id74149"/>
																			<m:apply id="id74150">
																				<m:divide id="id74151"/>
																				<m:ci id="id74152"/>
																				<m:ci id="id74153">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id74156">
																		<m:times id="id74157"/>
																		<m:apply id="id74158">
																			<m:csymbol id="id74159" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id74163">κ</m:ci>
																			<m:ci id="id74166">G</m:ci>
																		</m:apply>
																		<m:ci id="id74168">a</m:ci>
																	</m:apply>
																</m:apply>
																<m:eq id="id74170"/>
																<m:apply id="id74171">
																	<m:times id="id74172"/>
																	<m:apply id="id74173">
																		<m:divide id="id74174"/>
																		<m:cn id="id74175">1</m:cn>
																		<m:apply id="id74178">
																			<m:abs id="id74179"/>
																			<m:ci id="id74180">W</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id74182">
																		<m:apply id="id74183">
																			<m:csymbol id="id74184" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id74189"/>
																			<m:apply id="id74190">
																				<m:list id="id74191"/>
																				<m:ci id="id74192">X</m:ci>
																				<m:divide id="id74194"/>
																				<m:apply id="id74195">
																					<m:divide id="id74196"/>
																					<m:ci id="id74197"/>
																					<m:ci id="id74198">T</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id74200">
																			<m:times id="id74201"/>
																			<m:apply id="id74202">
																				<m:compose id="id74204"/>
																				<m:apply id="id74205">
																					<m:csymbol id="id74206" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id74210">κ</m:ci>
																					<m:ci id="id74213">T</m:ci>
																				</m:apply>
																				<m:apply id="id74215">
																					<m:csymbol id="id74216" cd="ambiguous">subscript</m:csymbol>
																					<m:apply id="id74221">
																						<m:csymbol id="id74222" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id74226">r</m:ci>
																						<m:ci id="id74228">G</m:ci>
																					</m:apply>
																					<m:ci id="id74231">T</m:ci>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id74233">a</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:ci id="id74235">⌣</m:ci>
																<m:ci id="id74237">e</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id74240" class="eqpad"/>
										</tr>

									</table>
									<p id="id74247" style="" class="p">
										<span id="id74252" style="" class="text italic">where</span>
									</p>
									<table class="equation" id="S3.Ex4">

										<tr valign="baseline" class="equation" id="S3.Ex4">
											<td id="id74275" class="eqpad"/>
											<td id="id74280" nowrap="yes" align="center" colspan="1">
												<m:math id="id74287" display="block">
													<m:semantics id="id74290">
														<m:mrow id="id74292">
															<m:mrow id="id74293">
																<m:mi id="id74294">e</m:mi>
																<m:mo id="id74296">=</m:mo>
																<m:mrow id="id74298">
																	<m:munder id="id74299">
																		<m:mo id="id74300" movablelimits="false">∏</m:mo>
																		<m:mrow id="id74305">
																			<m:mi id="id74306">α</m:mi>
																			<m:mo id="id74308">∈</m:mo>
																			<m:mi id="id74311" mathvariant="normal">Δ</m:mi>
																		</m:mrow>
																	</m:munder>
																	<m:msub id="id74315">
																		<m:mi id="id74316">κ</m:mi>
																		<m:mi id="id74319">T</m:mi>
																	</m:msub>
																	<m:mo id="id74321">⁢</m:mo>
																	<m:mfenced id="id74323" open="(" close=")">
																		<m:mrow id="id74328">
																			<m:msub id="id74329">
																				<m:mi id="id74330">e</m:mi>
																				<m:mi id="id74333">T</m:mi>
																			</m:msub>
																			<m:mo id="id74335">⁢</m:mo>
																			<m:mfenced id="id74337" open="(" close=")">
																				<m:msub id="id74342">
																					<m:mi id="id74343" mathvariant="double-struck">C</m:mi>
																					<m:mfenced id="id74348" open="(" close=")">
																						<m:mi id="id74353">α</m:mi>
																					</m:mfenced>
																				</m:msub>
																			</m:mfenced>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
															<m:mo id="id74355">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id74357" encoding="MathML-Content">
															<m:apply id="id74361">
																<m:eq id="id74362"/>
																<m:ci id="id74363">e</m:ci>
																<m:apply id="id74365">
																	<m:apply id="id74366">
																		<m:csymbol id="id74367" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id74372">∏</m:ci>
																		<m:apply id="id74374">
																			<m:ci id="id74375">∈</m:ci>
																			<m:ci id="id74378">α</m:ci>
																			<m:ci id="id74380">Δ</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id74382">
																		<m:times id="id74383"/>
																		<m:apply id="id74384">
																			<m:csymbol id="id74386" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id74390">κ</m:ci>
																			<m:ci id="id74393">T</m:ci>
																		</m:apply>
																		<m:apply id="id74395">
																			<m:times id="id74396"/>
																			<m:apply id="id74397">
																				<m:csymbol id="id74398" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id74403">e</m:ci>
																				<m:ci id="id74405">T</m:ci>
																			</m:apply>
																			<m:apply id="id74407">
																				<m:csymbol id="id74408" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id74413">C</m:ci>
																				<m:ci id="id74415">α</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id74418" class="eqpad"/>
										</tr>

									</table>
									<p id="id74425" style="" class="p">
										<span id="id74430" style="" class="text italic">and <m:math id="id74436" display="inline">
												<m:semantics id="id74440">
													<m:msub id="id74441">
														<m:mi id="id74442">e</m:mi>
														<m:mi id="id74444">T</m:mi>
													</m:msub>
													<m:annotation-xml id="id74446" encoding="MathML-Content">
														<m:apply id="id74450">
															<m:csymbol id="id74451" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id74455">e</m:ci>
															<m:ci id="id74457">T</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> denotes the <m:math id="id74461" display="inline">
												<m:semantics id="id74464">
													<m:mi id="id74465">T</m:mi>
													<m:annotation-xml id="id74467" encoding="MathML-Content">
														<m:ci id="id74471">T</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>-equivariant Euler class.</span>
									</p>
								</div>

							</div>

						</div>

						<div class="subsection" id="S3.SS2">
							<h3 id="id74485" class="title subsection-title">
								<span id="id74489" class="tag">3.2 </span>The proof of theorem A</h3>
							<div class="para" id="S3.SS2.p1">
								<p id="id74502" style="" class="p">Observe that the Weyl group <m:math id="id74509" display="inline">
										<m:semantics id="id74512">
											<m:mi id="id74513">W</m:mi>
											<m:annotation-xml id="id74515" encoding="MathML-Content">
												<m:ci id="id74519">W</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of <m:math id="id74522" display="inline">
										<m:semantics id="id74526">
											<m:mi id="id74527">G</m:mi>
											<m:annotation-xml id="id74529" encoding="MathML-Content">
												<m:ci id="id74532">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> acts on <m:math id="id74536" display="inline">
										<m:semantics id="id74539">
											<m:mrow id="id74540">
												<m:mi id="id74541">X</m:mi>
												<m:mo id="id74543">/</m:mo>
												<m:mo id="id74545">/</m:mo>
												<m:mi id="id74547">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id74550" encoding="MathML-Content">
												<m:ci id="id74553">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>: since the
normalizer <m:math id="id74556" display="inline">
										<m:semantics id="id74560">
											<m:mrow id="id74561">
												<m:mi id="id74562">N</m:mi>
												<m:mo id="id74564">⁢</m:mo>
												<m:mfenced id="id74566" open="(" close=")">
													<m:mi id="id74571">T</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id74574" encoding="MathML-Content">
												<m:apply id="id74577">
													<m:times id="id74578"/>
													<m:ci id="id74579">N</m:ci>
													<m:ci id="id74581">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of <m:math id="id74585" display="inline">
										<m:semantics id="id74588">
											<m:mi id="id74589">T</m:mi>
											<m:annotation-xml id="id74591" encoding="MathML-Content">
												<m:ci id="id74594">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> preserves <m:math id="id74598" display="inline">
										<m:semantics id="id74601">
											<m:mrow id="id74602">
												<m:msubsup id="id74603">
													<m:mi id="id74604">μ</m:mi>
													<m:mi id="id74607">T</m:mi>
													<m:mrow id="id74609">
														<m:mo id="id74610">-</m:mo>
														<m:mn id="id74612">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id74614">⁢</m:mo>
												<m:mfenced id="id74617" open="(" close=")">
													<m:mn id="id74622">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id74624" encoding="MathML-Content">
												<m:apply id="id74627">
													<m:times id="id74628"/>
													<m:apply id="id74629">
														<m:csymbol id="id74630" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id74635">
															<m:csymbol id="id74636" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id74641">μ</m:ci>
															<m:ci id="id74643">T</m:ci>
														</m:apply>
														<m:apply id="id74645">
															<m:minus id="id74646"/>
															<m:cn id="id74648">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id74650">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, the action of <m:math id="id74653" display="inline">
										<m:semantics id="id74656">
											<m:mrow id="id74658">
												<m:mi id="id74659">N</m:mi>
												<m:mo id="id74661">⁢</m:mo>
												<m:mfenced id="id74663" open="(" close=")">
													<m:mi id="id74668">T</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id74670" encoding="MathML-Content">
												<m:apply id="id74674">
													<m:times id="id74675"/>
													<m:ci id="id74676">N</m:ci>
													<m:ci id="id74678">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
on <m:math id="id74681" display="inline">
										<m:semantics id="id74685">
											<m:mrow id="id74686">
												<m:msubsup id="id74687">
													<m:mi id="id74688">μ</m:mi>
													<m:mi id="id74690">T</m:mi>
													<m:mrow id="id74692">
														<m:mo id="id74694">-</m:mo>
														<m:mn id="id74696">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id74698">⁢</m:mo>
												<m:mfenced id="id74700" open="(" close=")">
													<m:mn id="id74705">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id74707" encoding="MathML-Content">
												<m:apply id="id74711">
													<m:times id="id74712"/>
													<m:apply id="id74713">
														<m:csymbol id="id74714" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id74719">
															<m:csymbol id="id74720" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id74724">μ</m:ci>
															<m:ci id="id74727">T</m:ci>
														</m:apply>
														<m:apply id="id74729">
															<m:minus id="id74730"/>
															<m:cn id="id74731">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id74733">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> descends to an action of <m:math id="id74737" display="inline">
										<m:semantics id="id74740">
											<m:mrow id="id74741">
												<m:mi id="id74742">W</m:mi>
												<m:mo id="id74744">=</m:mo>
												<m:mrow id="id74746">
													<m:mrow id="id74748">
														<m:mi id="id74749">N</m:mi>
														<m:mo id="id74751">⁢</m:mo>
														<m:mfenced id="id74753" open="(" close=")">
															<m:mi id="id74758">T</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id74760">/</m:mo>
													<m:mi id="id74762">T</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id74765" encoding="MathML-Content">
												<m:apply id="id74768">
													<m:eq id="id74769"/>
													<m:ci id="id74770">W</m:ci>
													<m:apply id="id74772">
														<m:divide id="id74773"/>
														<m:apply id="id74774">
															<m:times id="id74775"/>
															<m:ci id="id74776">N</m:ci>
															<m:ci id="id74779">T</m:ci>
														</m:apply>
														<m:ci id="id74781">T</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on the quotient
									<m:math id="id74784" display="inline">
										<m:semantics id="id74788">
											<m:mrow id="id74789">
												<m:mi id="id74790">X</m:mi>
												<m:mo id="id74792">/</m:mo>
												<m:mo id="id74794">/</m:mo>
												<m:mi id="id74796">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id74798" encoding="MathML-Content">
												<m:ci id="id74802">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div id="id74807" class="proof">
								<h6 id="id74811" class="title proof-title italic">Proof of theorem A.</h6>
								<div class="para" id="S3.SS2.p2">
									<p id="id74824" style="" class="p">Consider the fibre bundle <m:math id="id74831" display="inline">
											<m:semantics id="id74834">
												<m:mrow id="id74835">
													<m:msubsup id="id74836">
														<m:mi id="id74837">μ</m:mi>
														<m:mi id="id74840">G</m:mi>
														<m:mrow id="id74842">
															<m:mo id="id74843">-</m:mo>
															<m:mn id="id74845">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id74847">(</m:mo>
													<m:mn id="id74849">0</m:mn>
													<m:mo id="id74851">)</m:mo>
													<m:mo id="id74853">/</m:mo>
													<m:mi id="id74856">T</m:mi>
													<m:mstyle id="id74858" scriptlevel="+1">
														<m:mtable id="id74861" rowspacing="0.2ex" columnspacing="0.4em">
															<m:mtr id="id74866">
																<m:mtd id="id74867">
																	<m:mi id="id74868">π</m:mi>
																</m:mtd>
															</m:mtr>
															<m:mtr id="id74871">
																<m:mtd id="id74872">
																	<m:mi id="id74873">→</m:mi>
																</m:mtd>
															</m:mtr>
														</m:mtable>
													</m:mstyle>
													<m:mi id="id74875">X</m:mi>
													<m:mo id="id74878">/</m:mo>
													<m:mo id="id74880">/</m:mo>
													<m:mi id="id74882">G</m:mi>
												</m:mrow>
												<m:annotation-xml id="id74884" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>. This has fibre
										<m:math id="id74889" display="inline">
											<m:semantics id="id74892">
												<m:mrow id="id74893">
													<m:mi id="id74894">G</m:mi>
													<m:mo id="id74896">/</m:mo>
													<m:mi id="id74898">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id74900" encoding="MathML-Content">
													<m:apply id="id74904">
														<m:divide id="id74905"/>
														<m:ci id="id74906">G</m:ci>
														<m:ci id="id74908">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and the Weyl group <m:math id="id74912" display="inline">
											<m:semantics id="id74915">
												<m:mi id="id74916">W</m:mi>
												<m:annotation-xml id="id74918" encoding="MathML-Content">
													<m:ci id="id74921">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> acts on the fibres, covering the trivial
action on the base (this is the restriction of the <m:math id="id74926" display="inline">
											<m:semantics id="id74929">
												<m:mi id="id74930">W</m:mi>
												<m:annotation-xml id="id74932" encoding="MathML-Content">
													<m:ci id="id74935">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-action on
										<m:math id="id74939" display="inline">
											<m:semantics id="id74942">
												<m:mrow id="id74943">
													<m:mi id="id74944">X</m:mi>
													<m:mo id="id74946">/</m:mo>
													<m:mo id="id74949">/</m:mo>
													<m:mi id="id74951">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id74953" encoding="MathML-Content">
													<m:ci id="id74956">X</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>). By a result of Borel <cite id="id74960" class="cite">[<a id="id74964" href="#bib.bib2" title="" class="ref">2</a>, section 27]</cite>, the
pullback <m:math id="id74975" display="inline">
											<m:semantics id="id74979">
												<m:msup id="id74980">
													<m:mi id="id74981">π</m:mi>
													<m:mo id="id74983">*</m:mo>
												</m:msup>
												<m:annotation-xml id="id74985" encoding="MathML-Content">
													<m:apply id="id74989">
														<m:csymbol id="id74990" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id74994">π</m:ci>
														<m:times id="id74997"/>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> gives an isomorphism between the rational cohomology
of the base <m:math id="id75000" display="inline">
											<m:semantics id="id75003">
												<m:mrow id="id75004">
													<m:mi id="id75005">X</m:mi>
													<m:mo id="id75007">/</m:mo>
													<m:mo id="id75009">/</m:mo>
													<m:mi id="id75012">G</m:mi>
												</m:mrow>
												<m:annotation-xml id="id75014" encoding="MathML-Content">
													<m:ci id="id75017">X</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> and the <m:math id="id75020" display="inline">
											<m:semantics id="id75024">
												<m:mi id="id75025">W</m:mi>
												<m:annotation-xml id="id75027" encoding="MathML-Content">
													<m:ci id="id75030">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-invariant cohomology of the total
space <m:math id="id75034" display="inline">
											<m:semantics id="id75037">
												<m:mrow id="id75038">
													<m:mrow id="id75040">
														<m:msubsup id="id75041">
															<m:mi id="id75042">μ</m:mi>
															<m:mi id="id75044">G</m:mi>
															<m:mrow id="id75046">
																<m:mo id="id75047">-</m:mo>
																<m:mn id="id75049">1</m:mn>
															</m:mrow>
														</m:msubsup>
														<m:mo id="id75052">⁢</m:mo>
														<m:mfenced id="id75054" open="(" close=")">
															<m:mn id="id75059">0</m:mn>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id75061">/</m:mo>
													<m:mi id="id75063">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id75065" encoding="MathML-Content">
													<m:apply id="id75069">
														<m:divide id="id75070"/>
														<m:apply id="id75071">
															<m:times id="id75072"/>
															<m:apply id="id75073">
																<m:csymbol id="id75074" cd="ambiguous">superscript</m:csymbol>
																<m:apply id="id75079">
																	<m:csymbol id="id75080" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id75084">μ</m:ci>
																	<m:ci id="id75087">G</m:ci>
																</m:apply>
																<m:apply id="id75089">
																	<m:minus id="id75090"/>
																	<m:cn id="id75091">1</m:cn>
																</m:apply>
															</m:apply>
															<m:cn id="id75093">0</m:cn>
														</m:apply>
														<m:ci id="id75095">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.  This means there is a natural ring homomorphism</p>
									<table class="equation" id="S3.E2">

										<tr valign="baseline" class="equation" id="S3.E2">
											<td id="id75116" class="eqpad"/>
											<td id="id75120" nowrap="yes" align="center" colspan="1">
												<m:math id="id75128" display="block">
													<m:semantics id="id75131">
														<m:mrow id="id75132">
															<m:mi id="id75133">φ</m:mi>
															<m:mo id="id75135">:</m:mo>
															<m:msup id="id75138">
																<m:mi id="id75139">H</m:mi>
																<m:mo id="id75141">*</m:mo>
															</m:msup>
															<m:mo id="id75143">(</m:mo>
															<m:mi id="id75145">X</m:mi>
															<m:mo id="id75147">/</m:mo>
															<m:mo id="id75149">/</m:mo>
															<m:mi id="id75151">T</m:mi>
															<m:msup id="id75154">
																<m:mo id="id75155">)</m:mo>
																<m:mi id="id75157">W</m:mi>
															</m:msup>
															<m:mo id="id75159">→</m:mo>
															<m:msup id="id75161">
																<m:mi id="id75162">H</m:mi>
																<m:mo id="id75164">*</m:mo>
															</m:msup>
															<m:mo id="id75167">(</m:mo>
															<m:mi id="id75169">X</m:mi>
															<m:mo id="id75171">/</m:mo>
															<m:mo id="id75173">/</m:mo>
															<m:mi id="id75175">G</m:mi>
															<m:mo id="id75177">)</m:mo>
														</m:mrow>
														<m:annotation-xml id="id75179" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id75184" class="eqpad"/>
											<td id="id75187" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id75198" class="refnum">3.2</span>)</td>
										</tr>

									</table>
									<p id="id75207" style="" class="p">given by restriction to <m:math id="id75213" display="inline">
											<m:semantics id="id75217">
												<m:mrow id="id75218">
													<m:mrow id="id75219">
														<m:msubsup id="id75220">
															<m:mi id="id75221">μ</m:mi>
															<m:mi id="id75223">G</m:mi>
															<m:mrow id="id75226">
																<m:mo id="id75227">-</m:mo>
																<m:mn id="id75229">1</m:mn>
															</m:mrow>
														</m:msubsup>
														<m:mo id="id75231">⁢</m:mo>
														<m:mfenced id="id75233" open="(" close=")">
															<m:mn id="id75238">0</m:mn>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id75240">/</m:mo>
													<m:mi id="id75243">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id75245" encoding="MathML-Content">
													<m:apply id="id75248">
														<m:divide id="id75249"/>
														<m:apply id="id75250">
															<m:times id="id75251"/>
															<m:apply id="id75252">
																<m:csymbol id="id75253" cd="ambiguous">superscript</m:csymbol>
																<m:apply id="id75258">
																	<m:csymbol id="id75259" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id75264">μ</m:ci>
																	<m:ci id="id75266">G</m:ci>
																</m:apply>
																<m:apply id="id75268">
																	<m:minus id="id75269"/>
																	<m:cn id="id75270">1</m:cn>
																</m:apply>
															</m:apply>
															<m:cn id="id75273">0</m:cn>
														</m:apply>
														<m:ci id="id75275">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> followed by the above identification,
which we will now show to be onto.</p>
								</div>

								<div class="para" id="S3.SS2.p3">
									<p id="id75288" style="" class="p">Applying the Borel-Hirzebruch result to the homotopy quotients <m:math id="id75294" display="inline">
											<m:semantics id="id75298">
												<m:msub id="id75299">
													<m:mi id="id75300">X</m:mi>
													<m:mi id="id75302">G</m:mi>
												</m:msub>
												<m:annotation-xml id="id75304" encoding="MathML-Content">
													<m:apply id="id75307">
														<m:csymbol id="id75308" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id75313">X</m:ci>
														<m:ci id="id75315">G</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
and <m:math id="id75319" display="inline">
											<m:semantics id="id75322">
												<m:msub id="id75323">
													<m:mi id="id75324">X</m:mi>
													<m:mi id="id75326">T</m:mi>
												</m:msub>
												<m:annotation-xml id="id75328" encoding="MathML-Content">
													<m:apply id="id75332">
														<m:csymbol id="id75333" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id75338">X</m:ci>
														<m:ci id="id75340">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, one can also recover the known fact that the restriction
										<m:math id="id75344" display="inline">
											<m:semantics id="id75347">
												<m:msubsup id="id75348">
													<m:mi id="id75349">r</m:mi>
													<m:mi id="id75351">T</m:mi>
													<m:mi id="id75353">G</m:mi>
												</m:msubsup>
												<m:annotation-xml id="id75355" encoding="MathML-Content">
													<m:apply id="id75359">
														<m:csymbol id="id75360" cd="ambiguous">subscript</m:csymbol>
														<m:apply id="id75364">
															<m:csymbol id="id75366" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id75370">r</m:ci>
															<m:ci id="id75372">G</m:ci>
														</m:apply>
														<m:ci id="id75374">T</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> gives an isomorphism with the
										<m:math id="id75378" display="inline">
											<m:semantics id="id75381">
												<m:mi id="id75382">W</m:mi>
												<m:annotation-xml id="id75385" encoding="MathML-Content">
													<m:ci id="id75388">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-invariants <m:math id="id75391" display="inline">
											<m:semantics id="id75395">
												<m:mrow id="id75396">
													<m:msubsup id="id75397">
														<m:mi id="id75398">r</m:mi>
														<m:mi id="id75400">T</m:mi>
														<m:mi id="id75402">G</m:mi>
													</m:msubsup>
													<m:mo id="id75404">:</m:mo>
													<m:mrow id="id75406">
														<m:mrow id="id75408">
															<m:msubsup id="id75409">
																<m:mi id="id75410">H</m:mi>
																<m:mi id="id75412">G</m:mi>
																<m:mo id="id75414">*</m:mo>
															</m:msubsup>
															<m:mo id="id75416">⁢</m:mo>
															<m:mfenced id="id75418" open="(" close=")">
																<m:mi id="id75424">X</m:mi>
															</m:mfenced>
														</m:mrow>
														<m:mstyle id="id75426" scriptlevel="+1">
															<m:mtable id="id75429" rowspacing="0.2ex" columnspacing="0.4em">
																<m:mtr id="id75434">
																	<m:mtd id="id75435">
																		<m:mo id="id75436">≅</m:mo>
																	</m:mtd>
																</m:mtr>
																<m:mtr id="id75439">
																	<m:mtd id="id75440">
																		<m:mi id="id75441">→</m:mi>
																	</m:mtd>
																</m:mtr>
															</m:mtable>
														</m:mstyle>
														<m:mrow id="id75443">
															<m:msubsup id="id75444">
																<m:mi id="id75446">H</m:mi>
																<m:mi id="id75448">T</m:mi>
																<m:mo id="id75450">*</m:mo>
															</m:msubsup>
															<m:mo id="id75452">⁢</m:mo>
															<m:msup id="id75454">
																<m:mfenced id="id75455" open="(" close=")">
																	<m:mi id="id75460">X</m:mi>
																</m:mfenced>
																<m:mi id="id75463">W</m:mi>
															</m:msup>
														</m:mrow>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id75465" encoding="MathML-Content">
													<m:apply id="id75468">
														<m:ci id="id75469">:</m:ci>
														<m:apply id="id75471">
															<m:csymbol id="id75472" cd="ambiguous">subscript</m:csymbol>
															<m:apply id="id75477">
																<m:csymbol id="id75478" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id75483">r</m:ci>
																<m:ci id="id75485">G</m:ci>
															</m:apply>
															<m:ci id="id75487">T</m:ci>
														</m:apply>
														<m:apply id="id75489">
															<m:apply id="id75490">
																<m:ci id="id75491"/>
																<m:approx id="id75492"/>
																<m:ci id="id75493">→</m:ci>
															</m:apply>
															<m:apply id="id75496">
																<m:times id="id75497"/>
																<m:apply id="id75498">
																	<m:csymbol id="id75499" cd="ambiguous">superscript</m:csymbol>
																	<m:apply id="id75504">
																		<m:csymbol id="id75505" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id75509">H</m:ci>
																		<m:ci id="id75512">G</m:ci>
																	</m:apply>
																	<m:times id="id75514"/>
																</m:apply>
																<m:ci id="id75515">X</m:ci>
															</m:apply>
															<m:apply id="id75517">
																<m:times id="id75518"/>
																<m:apply id="id75519">
																	<m:csymbol id="id75520" cd="ambiguous">superscript</m:csymbol>
																	<m:apply id="id75525">
																		<m:csymbol id="id75526" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id75530">H</m:ci>
																		<m:ci id="id75533">T</m:ci>
																	</m:apply>
																	<m:times id="id75535"/>
																</m:apply>
																<m:apply id="id75536">
																	<m:csymbol id="id75537" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id75542">X</m:ci>
																	<m:ci id="id75544">W</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.
By naturality of the maps involved, we have</p>
									<table class="equation" id="S3.Ex5">

										<tr valign="baseline" class="equation" id="S3.Ex5">
											<td id="id75564" class="eqpad"/>
											<td id="id75568" nowrap="yes" align="center" colspan="1">
												<m:math id="id75576" display="block">
													<m:semantics id="id75579">
														<m:mrow id="id75580">
															<m:msub id="id75581">
																<m:mi id="id75582">κ</m:mi>
																<m:mi id="id75585">G</m:mi>
															</m:msub>
															<m:mo id="id75587">=</m:mo>
															<m:mi id="id75589">φ</m:mi>
															<m:mo id="id75591">∘</m:mo>
															<m:msub id="id75594">
																<m:mi id="id75595">κ</m:mi>
																<m:mi id="id75597">T</m:mi>
															</m:msub>
															<m:mo id="id75599">∘</m:mo>
															<m:msubsup id="id75602">
																<m:mi id="id75603">r</m:mi>
																<m:mi id="id75605">T</m:mi>
																<m:mi id="id75607">G</m:mi>
															</m:msubsup>
															<m:mo id="id75609">:</m:mo>
															<m:msubsup id="id75611">
																<m:mi id="id75612">H</m:mi>
																<m:mi id="id75615">G</m:mi>
																<m:mo id="id75617">*</m:mo>
															</m:msubsup>
															<m:mo id="id75619">(</m:mo>
															<m:mi id="id75621">X</m:mi>
															<m:mo id="id75623">)</m:mo>
															<m:mo id="id75625">→</m:mo>
															<m:msup id="id75628">
																<m:mi id="id75629">H</m:mi>
																<m:mo id="id75631">*</m:mo>
															</m:msup>
															<m:mo id="id75633">(</m:mo>
															<m:mi id="id75635">X</m:mi>
															<m:mo id="id75637">/</m:mo>
															<m:mo id="id75639">/</m:mo>
															<m:mi id="id75642">G</m:mi>
															<m:mo id="id75644">)</m:mo>
															<m:mo id="id75646">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id75648" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id75652" class="eqpad"/>
										</tr>

									</table>
									<p id="id75659" style="" class="p">and since <m:math id="id75665" display="inline">
											<m:semantics id="id75669">
												<m:msub id="id75670">
													<m:mi id="id75671">κ</m:mi>
													<m:mi id="id75673">G</m:mi>
												</m:msub>
												<m:annotation-xml id="id75675" encoding="MathML-Content">
													<m:apply id="id75679">
														<m:csymbol id="id75680" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id75684">κ</m:ci>
														<m:ci id="id75687">G</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is onto, it follows that <m:math id="id75690" display="inline">
											<m:semantics id="id75694">
												<m:mi id="id75695">φ</m:mi>
												<m:annotation-xml id="id75697" encoding="MathML-Content">
													<m:ci id="id75700">φ</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is onto.</p>
								</div>

								<div class="para" id="S3.SS2.p4">
									<p id="id75713" style="" class="p">To prove the theorem,  we thus need to show that <m:math id="id75720" display="inline">
											<m:semantics id="id75723">
												<m:mrow id="id75724">
													<m:mrow id="id75725">
														<m:mi id="id75726">ker</m:mi>
														<m:mo id="id75728">⁡</m:mo>
														<m:mi id="id75731">φ</m:mi>
													</m:mrow>
													<m:mo id="id75733">=</m:mo>
													<m:mrow id="id75735">
														<m:mo id="id75736">ann</m:mo>
														<m:mo id="id75738">⁡</m:mo>
														<m:mfenced id="id75741" open="(" close=")">
															<m:mi id="id75746">e</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:annotation-xml id="id75748" encoding="MathML-Content">
													<m:apply id="id75751">
														<m:eq id="id75752"/>
														<m:apply id="id75754">
															<m:ci id="id75755">ker</m:ci>
															<m:ci id="id75757">φ</m:ci>
														</m:apply>
														<m:apply id="id75759">
															<m:ci id="id75760">ann</m:ci>
															<m:ci id="id75762">e</m:ci>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.
Let
										<m:math id="id75766" display="inline">
											<m:semantics id="id75769">
												<m:mrow id="id75770">
													<m:mi id="id75771">a</m:mi>
													<m:mo id="id75773">∈</m:mo>
													<m:msup id="id75776">
														<m:mi id="id75777">H</m:mi>
														<m:mo id="id75779">*</m:mo>
													</m:msup>
													<m:mo id="id75781">(</m:mo>
													<m:mi id="id75783">X</m:mi>
													<m:mo id="id75785">/</m:mo>
													<m:mo id="id75788">/</m:mo>
													<m:mi id="id75790">T</m:mi>
													<m:msup id="id75792">
														<m:mo id="id75793">)</m:mo>
														<m:mi id="id75795">W</m:mi>
													</m:msup>
												</m:mrow>
												<m:annotation-xml id="id75797" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>. Then</p>
									<table class="equation" id="S3.Ex6">

										<tr valign="baseline" class="equation" id="S3.Ex6">
											<td id="id75818" class="eqpad"/>
											<td id="id75823" nowrap="yes" align="center" colspan="1">
												<m:math id="id75830" display="block">
													<m:semantics id="id75834">
														<m:mrow id="id75835">
															<m:mi id="id75836">φ</m:mi>
															<m:mo id="id75838">(</m:mo>
															<m:mi id="id75840">a</m:mi>
															<m:mo id="id75842">)</m:mo>
															<m:mo id="id75844">=</m:mo>
															<m:mn id="id75847">0</m:mn>
															<m:mo id="id75849">⇔</m:mo>
															<m:mi id="id75851" mathvariant="normal">∀</m:mi>
															<m:mi id="id75856">c</m:mi>
															<m:mo id="id75858">∈</m:mo>
															<m:msup id="id75860">
																<m:mi id="id75861">H</m:mi>
																<m:mo id="id75864">*</m:mo>
															</m:msup>
															<m:mo id="id75866">(</m:mo>
															<m:mi id="id75868">X</m:mi>
															<m:mo id="id75870">/</m:mo>
															<m:mo id="id75872">/</m:mo>
															<m:mi id="id75874">T</m:mi>
															<m:msup id="id75876">
																<m:mo id="id75877">)</m:mo>
																<m:mi id="id75880">W</m:mi>
															</m:msup>
															<m:mo id="id75882">,</m:mo>
															<m:msub id="id75884">
																<m:mo id="id75885">∫</m:mo>
																<m:mrow id="id75887">
																	<m:mi id="id75888">X</m:mi>
																	<m:mo id="id75890">⁣</m:mo>
																	<m:mo id="id75893">/</m:mo>
																	<m:mo id="id75895">⁣</m:mo>
																	<m:mrow id="id75897">
																		<m:none id="id75898"/>
																		<m:mo id="id75900">/</m:mo>
																		<m:mi id="id75902">G</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mi id="id75904">φ</m:mi>
															<m:mo id="id75906">(</m:mo>
															<m:mi id="id75908">a</m:mi>
															<m:mo id="id75910">)</m:mo>
															<m:mo id="id75913">⌣</m:mo>
															<m:mi id="id75915">φ</m:mi>
															<m:mo id="id75917">(</m:mo>
															<m:mi id="id75920">c</m:mi>
															<m:mo id="id75922">)</m:mo>
															<m:mo id="id75924">=</m:mo>
															<m:mn id="id75926">0</m:mn>
															<m:mo id="id75928">,</m:mo>
															<m:mrow id="id75930">
																<m:mtext id="id75931">Poincaré duality on </m:mtext>
																<m:mrow id="id75934">
																	<m:mi id="id75935">X</m:mi>
																	<m:mo id="id75937">/</m:mo>
																	<m:mo id="id75939">/</m:mo>
																	<m:mi id="id75941">G</m:mi>
																</m:mrow>
																<m:mtext id="id75944">;</m:mtext>
																<m:mtext id="id75946"> surjectivity of </m:mtext>
																<m:mi id="id75948">φ</m:mi>
																<m:mtext id="id75950">,</m:mtext>
															</m:mrow>
															<m:mo id="id75953">⇔</m:mo>
															<m:mi id="id75955" mathvariant="normal">∀</m:mi>
															<m:mi id="id75960">c</m:mi>
															<m:mo id="id75962">∈</m:mo>
															<m:msup id="id75964">
																<m:mi id="id75965">H</m:mi>
																<m:mo id="id75967">*</m:mo>
															</m:msup>
															<m:mo id="id75970">(</m:mo>
															<m:mi id="id75972">X</m:mi>
															<m:mo id="id75974">/</m:mo>
															<m:mo id="id75976">/</m:mo>
															<m:mi id="id75978">T</m:mi>
															<m:msup id="id75980">
																<m:mo id="id75981">)</m:mo>
																<m:mi id="id75983">W</m:mi>
															</m:msup>
															<m:mo id="id75986">,</m:mo>
															<m:msub id="id75988">
																<m:mo id="id75989">∫</m:mo>
																<m:mrow id="id75991">
																	<m:mi id="id75992">X</m:mi>
																	<m:mo id="id75994">⁣</m:mo>
																	<m:mo id="id75997">/</m:mo>
																	<m:mo id="id75999">⁣</m:mo>
																	<m:mrow id="id76001">
																		<m:none id="id76002"/>
																		<m:mo id="id76003">/</m:mo>
																		<m:mi id="id76006">T</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mi id="id76008">a</m:mi>
															<m:mo id="id76010">⌣</m:mo>
															<m:mi id="id76012">c</m:mi>
															<m:mo id="id76014">⌣</m:mo>
															<m:mi id="id76017">e</m:mi>
															<m:mo id="id76019">=</m:mo>
															<m:mn id="id76021">0</m:mn>
															<m:mo id="id76023">,</m:mo>
															<m:mtext id="id76025">by the integration formula,</m:mtext>
															<m:mo id="id76028">⇔</m:mo>
															<m:mi id="id76030" mathvariant="normal">∀</m:mi>
															<m:mi id="id76035">d</m:mi>
															<m:mo id="id76037">∈</m:mo>
															<m:msup id="id76039">
																<m:mi id="id76040">H</m:mi>
																<m:mo id="id76043">*</m:mo>
															</m:msup>
															<m:mo id="id76045">(</m:mo>
															<m:mi id="id76047">X</m:mi>
															<m:mo id="id76049">/</m:mo>
															<m:mo id="id76051">/</m:mo>
															<m:mi id="id76053">T</m:mi>
															<m:mo id="id76055">)</m:mo>
															<m:mo id="id76058">,</m:mo>
															<m:msub id="id76060">
																<m:mo id="id76061">∫</m:mo>
																<m:mrow id="id76063">
																	<m:mi id="id76064">X</m:mi>
																	<m:mo id="id76066">⁣</m:mo>
																	<m:mo id="id76069">/</m:mo>
																	<m:mo id="id76071">⁣</m:mo>
																	<m:mrow id="id76073">
																		<m:none id="id76074"/>
																		<m:mo id="id76075">/</m:mo>
																		<m:mi id="id76078">T</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:msub>
															<m:mo id="id76080">(</m:mo>
															<m:mi id="id76082">a</m:mi>
															<m:mo id="id76084">⌣</m:mo>
															<m:mi id="id76086">e</m:mi>
															<m:mo id="id76088">)</m:mo>
															<m:mo id="id76091">⌣</m:mo>
															<m:mi id="id76093">d</m:mi>
															<m:mo id="id76095">=</m:mo>
															<m:mn id="id76097">0</m:mn>
															<m:mo id="id76099">,</m:mo>
															<m:mrow id="id76102">
																<m:mtext id="id76103">since we can average </m:mtext>
																<m:mi id="id76105">d</m:mi>
																<m:mtext id="id76107"> by </m:mtext>
																<m:mi id="id76110">W</m:mi>
																<m:mtext id="id76112"> (see below),</m:mtext>
															</m:mrow>
															<m:mo id="id76114">⇔</m:mo>
															<m:mi id="id76117">a</m:mi>
															<m:mo id="id76119">⌣</m:mo>
															<m:mi id="id76121">e</m:mi>
															<m:mo id="id76123">=</m:mo>
															<m:mn id="id76126">0</m:mn>
															<m:mo id="id76128">,</m:mo>
															<m:mrow id="id76130">
																<m:mtext id="id76131">by Poincaré duality on </m:mtext>
																<m:mrow id="id76134">
																	<m:mi id="id76135">X</m:mi>
																	<m:mo id="id76137">/</m:mo>
																	<m:mo id="id76139">/</m:mo>
																	<m:mi id="id76141">T</m:mi>
																</m:mrow>
																<m:mtext id="id76143">.</m:mtext>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id76145" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id76150" class="eqpad"/>
										</tr>

									</table>
									<p id="id76156" style="" class="p">In the second-last step, we note that <m:math id="id76163" display="inline">
											<m:semantics id="id76166">
												<m:mi id="id76167">W</m:mi>
												<m:annotation-xml id="id76169" encoding="MathML-Content">
													<m:ci id="id76173">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> acts by symplectomorphisms
on <m:math id="id76176" display="inline">
											<m:semantics id="id76180">
												<m:mrow id="id76181">
													<m:mi id="id76182">X</m:mi>
													<m:mo id="id76184">/</m:mo>
													<m:mo id="id76186">/</m:mo>
													<m:mi id="id76188">T</m:mi>
												</m:mrow>
												<m:annotation-xml id="id76190" encoding="MathML-Content">
													<m:ci id="id76194">X</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, hence preserves orientation and integrals, and since <m:math id="id76197" display="inline">
											<m:semantics id="id76201">
												<m:mi id="id76202">a</m:mi>
												<m:annotation-xml id="id76204" encoding="MathML-Content">
													<m:ci id="id76207">a</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
and <m:math id="id76211" display="inline">
											<m:semantics id="id76214">
												<m:mi id="id76215">e</m:mi>
												<m:annotation-xml id="id76217" encoding="MathML-Content">
													<m:ci id="id76220">e</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> are <m:math id="id76224" display="inline">
											<m:semantics id="id76227">
												<m:mi id="id76228">W</m:mi>
												<m:annotation-xml id="id76230" encoding="MathML-Content">
													<m:ci id="id76234">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-invariant, we can average <m:math id="id76237" display="inline">
											<m:semantics id="id76241">
												<m:mi id="id76242">d</m:mi>
												<m:annotation-xml id="id76244" encoding="MathML-Content">
													<m:ci id="id76247">d</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> by <m:math id="id76251" display="inline">
											<m:semantics id="id76254">
												<m:mi id="id76255">W</m:mi>
												<m:annotation-xml id="id76257" encoding="MathML-Content">
													<m:ci id="id76261">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> to obtain a
										<m:math id="id76264" display="inline">
											<m:semantics id="id76267">
												<m:mi id="id76268">W</m:mi>
												<m:annotation-xml id="id76271" encoding="MathML-Content">
													<m:ci id="id76274">W</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-invariant class without changing the integral.
∎</p>
								</div>

							</div>

						</div>

					</div>

					<div class="section" id="S4">
						<h2 id="id76290" class="title section-title">
							<span id="id76294" class="tag">4 </span>The index formula</h2>
						<div class="para" id="S4.p1">
							<p id="id76307" style="" class="p">We now prove the index formula, theorem C, by applying the
Atiyah-Singer index theorem to the main topological result.
For more details on the spinc<m:math id="id76316" display="inline">
									<m:semantics id="id76319">
										<m:msup id="id76320">
											<m:mi id="id76321"/>
											<m:mi id="id76322">c</m:mi>
										</m:msup>
										<m:annotation-xml id="id76324" encoding="MathML-Content">
											<m:apply id="id76328">
												<m:ci id="id76329">c</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> Dirac
operators <m:math id="id76332" display="inline">
									<m:semantics id="id76335">
										<m:mi id="id76336">D</m:mi>
										<m:annotation-xml id="id76339" encoding="MathML-Content">
											<m:ci id="id76342">D</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id76345" display="inline">
									<m:semantics id="id76349">
										<m:msub id="id76350">
											<m:mi id="id76351">D</m:mi>
											<m:mi id="id76353">V</m:mi>
										</m:msub>
										<m:annotation-xml id="id76355" encoding="MathML-Content">
											<m:apply id="id76358">
												<m:csymbol id="id76360" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id76364">D</m:ci>
												<m:ci id="id76366">V</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> (described in the introduction) see for
example <cite id="id76370" class="cite">[<a id="id76375" href="#bib.bib14" title="" class="ref">14</a>, appendix D]</cite>.</p>
						</div>

						<div class="para" id="S4.p2">
							<p id="id76394" style="" class="p">In the proof we will use <m:math id="id76401" display="inline">
									<m:semantics id="id76404">
										<m:mi id="id76405">K</m:mi>
										<m:annotation-xml id="id76408" encoding="MathML-Content">
											<m:ci id="id76411">K</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-theory, but only in a rudimentary way,
and we recall a few facts and set up some notation.  Given a compact
space <m:math id="id76415" display="inline">
									<m:semantics id="id76418">
										<m:mi id="id76420">Y</m:mi>
										<m:annotation-xml id="id76422" encoding="MathML-Content">
											<m:ci id="id76425">Y</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, then <m:math id="id76428" display="inline">
									<m:semantics id="id76432">
										<m:mrow id="id76433">
											<m:mi id="id76434">K</m:mi>
											<m:mo id="id76436">⁢</m:mo>
											<m:mfenced id="id76438" open="(" close=")">
												<m:mi id="id76444">Y</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id76446" encoding="MathML-Content">
											<m:apply id="id76449">
												<m:times id="id76450"/>
												<m:ci id="id76451">K</m:ci>
												<m:ci id="id76453">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a commutative ring, whose elements are
represented by formal sums (with integer coefficients) of complex
vector bundles over <m:math id="id76458" display="inline">
									<m:semantics id="id76461">
										<m:mi id="id76462">Y</m:mi>
										<m:annotation-xml id="id76464" encoding="MathML-Content">
											<m:ci id="id76468">Y</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Given a vector bundle <m:math id="id76471" display="inline">
									<m:semantics id="id76474">
										<m:mrow id="id76476">
											<m:mi id="id76477">V</m:mi>
											<m:mo id="id76479">→</m:mo>
											<m:mi id="id76481">Y</m:mi>
										</m:mrow>
										<m:annotation-xml id="id76483" encoding="MathML-Content">
											<m:apply id="id76487">
												<m:ci id="id76488">→</m:ci>
												<m:ci id="id76490">V</m:ci>
												<m:ci id="id76492">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, we write
								<m:math id="id76496" display="inline">
									<m:semantics id="id76499">
										<m:mrow id="id76500">
											<m:mfenced id="id76501" open="[" close="]">
												<m:mi id="id76506">V</m:mi>
											</m:mfenced>
											<m:mo id="id76508">∈</m:mo>
											<m:mrow id="id76511">
												<m:mi id="id76512">K</m:mi>
												<m:mo id="id76514">⁢</m:mo>
												<m:mfenced id="id76516" open="(" close=")">
													<m:mi id="id76521">Y</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id76524" encoding="MathML-Content">
											<m:apply id="id76527">
												<m:ci id="id76528">∈</m:ci>
												<m:ci id="id76530">V</m:ci>
												<m:apply id="id76532">
													<m:times id="id76534"/>
													<m:ci id="id76535">K</m:ci>
													<m:ci id="id76537">Y</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> for the equivalence class it represents. Addition and
multiplication in <m:math id="id76541" display="inline">
									<m:semantics id="id76544">
										<m:mrow id="id76545">
											<m:mi id="id76546">K</m:mi>
											<m:mo id="id76548">⁢</m:mo>
											<m:mfenced id="id76551" open="(" close=")">
												<m:mi id="id76556">Y</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id76558" encoding="MathML-Content">
											<m:apply id="id76561">
												<m:times id="id76562"/>
												<m:ci id="id76563">K</m:ci>
												<m:ci id="id76566">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are induced by the direct sum and tensor
product of vector bundles respectively, and these operations are
extended to formal sums of vector bundles by the usual laws of a
commutative ring.
							</p>
						</div>

						<div class="para" id="S4.p3">
							<p id="id76579" style="" class="p">We will use the Chern character and the Todd class of complex vector
bundles. It is a standard fact that these characteristic classes only
depend on the equivalence class <m:math id="id76587" display="inline">
									<m:semantics id="id76590">
										<m:mrow id="id76591">
											<m:mfenced id="id76592" open="[" close="]">
												<m:mi id="id76598">V</m:mi>
											</m:mfenced>
											<m:mo id="id76600">∈</m:mo>
											<m:mrow id="id76602">
												<m:mi id="id76603">K</m:mi>
												<m:mo id="id76605">⁢</m:mo>
												<m:mfenced id="id76608" open="(" close=")">
													<m:mi id="id76613">Y</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id76615" encoding="MathML-Content">
											<m:apply id="id76618">
												<m:ci id="id76619">∈</m:ci>
												<m:ci id="id76622">V</m:ci>
												<m:apply id="id76624">
													<m:times id="id76625"/>
													<m:ci id="id76626">K</m:ci>
													<m:ci id="id76628">Y</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of a complex vector
bundle <m:math id="id76632" display="inline">
									<m:semantics id="id76635">
										<m:mrow id="id76636">
											<m:mi id="id76637">V</m:mi>
											<m:mo id="id76639">→</m:mo>
											<m:mi id="id76642">Y</m:mi>
										</m:mrow>
										<m:annotation-xml id="id76644" encoding="MathML-Content">
											<m:apply id="id76647">
												<m:ci id="id76648">→</m:ci>
												<m:ci id="id76651">V</m:ci>
												<m:ci id="id76653">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and that these characteristic classes can be extended
to every element of <m:math id="id76657" display="inline">
									<m:semantics id="id76660">
										<m:mrow id="id76661">
											<m:mi id="id76662">K</m:mi>
											<m:mo id="id76664">⁢</m:mo>
											<m:mfenced id="id76667" open="(" close=")">
												<m:mi id="id76672">Y</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id76674" encoding="MathML-Content">
											<m:apply id="id76677">
												<m:times id="id76678"/>
												<m:ci id="id76679">K</m:ci>
												<m:ci id="id76682">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by setting</p>
							<table class="equation" id="S4.Ex1">

								<tr valign="baseline" class="equation" id="S4.Ex1">
									<td id="id76702" class="eqpad"/>
									<td id="id76706" nowrap="yes" align="center" colspan="1">
										<m:math id="id76713" display="block">
											<m:semantics id="id76717">
												<m:mrow id="id76718">
													<m:mrow id="id76719">
														<m:mrow id="id76720">
															<m:mrow id="id76721">
																<m:mrow id="id76722">
																	<m:mi id="id76723">T</m:mi>
																	<m:mo id="id76725">⁢</m:mo>
																	<m:mi id="id76728">d</m:mi>
																</m:mrow>
																<m:mo id="id76730">⁢</m:mo>
																<m:mfenced id="id76732" open="(" close=")">
																	<m:mrow id="id76737">
																		<m:mfenced id="id76738" open="[" close="]">
																			<m:mi id="id76743">V</m:mi>
																		</m:mfenced>
																		<m:mo id="id76746">-</m:mo>
																		<m:mfenced id="id76748" open="[" close="]">
																			<m:mi id="id76753">W</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id76755">=</m:mo>
															<m:mfrac id="id76757">
																<m:mrow id="id76758">
																	<m:mrow id="id76759">
																		<m:mi id="id76760">T</m:mi>
																		<m:mo id="id76762">⁢</m:mo>
																		<m:mi id="id76765">d</m:mi>
																	</m:mrow>
																	<m:mo id="id76767">⁢</m:mo>
																	<m:mfenced id="id76769" open="(" close=")">
																		<m:mi id="id76774">V</m:mi>
																	</m:mfenced>
																</m:mrow>
																<m:mrow id="id76776">
																	<m:mrow id="id76778">
																		<m:mi id="id76779">T</m:mi>
																		<m:mo id="id76781">⁢</m:mo>
																		<m:mi id="id76783">d</m:mi>
																	</m:mrow>
																	<m:mo id="id76785">⁢</m:mo>
																	<m:mfenced id="id76788" open="(" close=")">
																		<m:mi id="id76793">W</m:mi>
																	</m:mfenced>
																</m:mrow>
															</m:mfrac>
														</m:mrow>
														<m:mo id="id76795">,</m:mo>
														<m:mrow id="id76797">
															<m:mrow id="id76798">
																<m:mrow id="id76799">
																	<m:mi id="id76800">c</m:mi>
																	<m:mo id="id76802">⁢</m:mo>
																	<m:mi id="id76805">h</m:mi>
																</m:mrow>
																<m:mo id="id76807">⁢</m:mo>
																<m:mfenced id="id76810" open="(" close=")">
																	<m:mrow id="id76815">
																		<m:mfenced id="id76816" open="[" close="]">
																			<m:mi id="id76821">V</m:mi>
																		</m:mfenced>
																		<m:mo id="id76823">-</m:mo>
																		<m:mfenced id="id76825" open="[" close="]">
																			<m:mi id="id76830">W</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id76832">=</m:mo>
															<m:mrow id="id76834">
																<m:mrow id="id76836">
																	<m:mrow id="id76837">
																		<m:mi id="id76838">c</m:mi>
																		<m:mo id="id76840">⁢</m:mo>
																		<m:mi id="id76842">h</m:mi>
																	</m:mrow>
																	<m:mo id="id76844">⁢</m:mo>
																	<m:mfenced id="id76847" open="(" close=")">
																		<m:mi id="id76852">V</m:mi>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id76854">-</m:mo>
																<m:mrow id="id76856">
																	<m:mrow id="id76857">
																		<m:mi id="id76858">c</m:mi>
																		<m:mo id="id76860">⁢</m:mo>
																		<m:mi id="id76863">h</m:mi>
																	</m:mrow>
																	<m:mo id="id76865">⁢</m:mo>
																	<m:mfenced id="id76867" open="(" close=")">
																		<m:mi id="id76872">W</m:mi>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo id="id76874">,</m:mo>
												</m:mrow>
												<m:annotation-xml id="id76877" encoding="MathML-Content">
													<m:apply id="id76880">
														<m:ci id="id76881"/>
														<m:apply id="id76882">
															<m:eq id="id76883"/>
															<m:apply id="id76884">
																<m:times id="id76885"/>
																<m:ci id="id76886"/>
																<m:apply id="id76887">
																	<m:minus id="id76888"/>
																	<m:ci id="id76890">V</m:ci>
																	<m:ci id="id76892">W</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id76894">
																<m:divide id="id76895"/>
																<m:apply id="id76896">
																	<m:times id="id76897"/>
																	<m:ci id="id76898"/>
																	<m:ci id="id76899">V</m:ci>
																</m:apply>
																<m:apply id="id76901">
																	<m:times id="id76902"/>
																	<m:ci id="id76903"/>
																	<m:ci id="id76904">W</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id76907">
															<m:eq id="id76908"/>
															<m:apply id="id76909">
																<m:times id="id76910"/>
																<m:ci id="id76911"/>
																<m:apply id="id76912">
																	<m:minus id="id76913"/>
																	<m:ci id="id76914">V</m:ci>
																	<m:ci id="id76916">W</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id76918">
																<m:minus id="id76919"/>
																<m:apply id="id76920">
																	<m:times id="id76922"/>
																	<m:ci id="id76923"/>
																	<m:ci id="id76924">V</m:ci>
																</m:apply>
																<m:apply id="id76926">
																	<m:times id="id76927"/>
																	<m:ci id="id76928"/>
																	<m:ci id="id76929">W</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td id="id76932" class="eqpad"/>
								</tr>

							</table>
							<p id="id76939" style="" class="p">for all vector bundles <m:math id="id76945" display="inline">
									<m:semantics id="id76948">
										<m:mrow id="id76950">
											<m:mrow id="id76951">
												<m:mi id="id76952">V</m:mi>
												<m:mo id="id76954">,</m:mo>
												<m:mi id="id76956">W</m:mi>
											</m:mrow>
											<m:mo id="id76958">→</m:mo>
											<m:mi id="id76960">Y</m:mi>
										</m:mrow>
										<m:annotation-xml id="id76963" encoding="MathML-Content">
											<m:apply id="id76966">
												<m:ci id="id76967">→</m:ci>
												<m:apply id="id76969">
													<m:list id="id76970"/>
													<m:ci id="id76972">V</m:ci>
													<m:ci id="id76974">W</m:ci>
												</m:apply>
												<m:ci id="id76976">Y</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> (the Todd class <m:math id="id76979" display="inline">
									<m:semantics id="id76983">
										<m:mrow id="id76984">
											<m:mrow id="id76985">
												<m:mi id="id76986">T</m:mi>
												<m:mo id="id76988">⁢</m:mo>
												<m:mi id="id76990">d</m:mi>
											</m:mrow>
											<m:mo id="id76992">⁢</m:mo>
											<m:mfenced id="id76995" open="(" close=")">
												<m:mi id="id77000">V</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id77002" encoding="MathML-Content">
											<m:apply id="id77005">
												<m:times id="id77006"/>
												<m:ci id="id77008"/>
												<m:ci id="id77009">V</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a
cohomology class of mixed degree, but it has degree-<m:math id="id77012" display="inline">
									<m:semantics id="id77016">
										<m:mn id="id77017">0</m:mn>
										<m:annotation-xml id="id77019" encoding="MathML-Content">
											<m:cn id="id77022">0</m:cn>
										</m:annotation-xml>
									</m:semantics>
								</m:math> part equal to
								<m:math id="id77026" display="inline">
									<m:semantics id="id77029">
										<m:mrow id="id77030">
											<m:mn id="id77031">1</m:mn>
											<m:mo id="id77033">∈</m:mo>
											<m:mrow id="id77036">
												<m:msup id="id77037">
													<m:mi id="id77038">H</m:mi>
													<m:mn id="id77040">0</m:mn>
												</m:msup>
												<m:mo id="id77042">⁢</m:mo>
												<m:mfenced id="id77045" open="(" close=")">
													<m:mi id="id77050">Y</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id77052" encoding="MathML-Content">
											<m:apply id="id77055">
												<m:ci id="id77056">∈</m:ci>
												<m:cn id="id77059">1</m:cn>
												<m:apply id="id77061">
													<m:times id="id77062"/>
													<m:apply id="id77063">
														<m:csymbol id="id77064" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id77069">H</m:ci>
														<m:cn id="id77071">0</m:cn>
													</m:apply>
													<m:ci id="id77073">Y</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and it follows that <m:math id="id77076" display="inline">
									<m:semantics id="id77080">
										<m:mrow id="id77081">
											<m:mrow id="id77082">
												<m:mi id="id77083">T</m:mi>
												<m:mo id="id77085">⁢</m:mo>
												<m:mi id="id77087">d</m:mi>
											</m:mrow>
											<m:mo id="id77090">⁢</m:mo>
											<m:mfenced id="id77092" open="(" close=")">
												<m:mi id="id77097">V</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id77099" encoding="MathML-Content">
											<m:apply id="id77102">
												<m:times id="id77104"/>
												<m:ci id="id77105"/>
												<m:ci id="id77106">V</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> has a multiplicative
inverse in the cohomology ring).  Finally, if a vector bundle <m:math id="id77110" display="inline">
									<m:semantics id="id77113">
										<m:mi id="id77114">V</m:mi>
										<m:annotation-xml id="id77116" encoding="MathML-Content">
											<m:ci id="id77120">V</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is
given as a sum of line bundles <m:math id="id77123" display="inline">
									<m:semantics id="id77126">
										<m:mrow id="id77128">
											<m:mi id="id77129">V</m:mi>
											<m:mo id="id77131">=</m:mo>
											<m:mrow id="id77133">
												<m:msub id="id77134">
													<m:mo id="id77135">⊕</m:mo>
													<m:mrow id="id77137">
														<m:mn id="id77138">1</m:mn>
														<m:mo id="id77141">≤</m:mo>
														<m:mi id="id77143">i</m:mi>
														<m:mo id="id77145">≤</m:mo>
														<m:mi id="id77148">k</m:mi>
													</m:mrow>
												</m:msub>
												<m:msub id="id77150">
													<m:mi id="id77151">L</m:mi>
													<m:mi id="id77153">i</m:mi>
												</m:msub>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id77155" encoding="MathML-Content">
											<m:apply id="id77158">
												<m:eq id="id77159"/>
												<m:ci id="id77160">V</m:ci>
												<m:apply id="id77163">
													<m:apply id="id77164">
														<m:csymbol id="id77165" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id77169">⊕</m:ci>
														<m:apply id="id77172">
															<m:ci id="id77173"/>
															<m:cn id="id77174">1</m:cn>
															<m:leq id="id77176"/>
															<m:ci id="id77177">i</m:ci>
															<m:leq id="id77179"/>
															<m:ci id="id77180">k</m:ci>
														</m:apply>
													</m:apply>
													<m:apply id="id77182">
														<m:csymbol id="id77184" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id77188">L</m:ci>
														<m:ci id="id77190">i</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> then</p>
							<table class="equation" id="S4.Ex2">

								<tr valign="baseline" class="equation" id="S4.Ex2">
									<td id="id77210" class="eqpad"/>
									<td id="id77215" nowrap="yes" align="center" colspan="1">
										<m:math id="id77222" display="block">
											<m:semantics id="id77226">
												<m:mrow id="id77227">
													<m:mrow id="id77228">
														<m:mrow id="id77229">
															<m:mrow id="id77230">
																<m:mrow id="id77231">
																	<m:mi id="id77232">T</m:mi>
																	<m:mo id="id77234">⁢</m:mo>
																	<m:mi id="id77236">d</m:mi>
																</m:mrow>
																<m:mo id="id77239">⁢</m:mo>
																<m:mfenced id="id77241" open="(" close=")">
																	<m:mi id="id77246">V</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id77248">=</m:mo>
															<m:mrow id="id77250">
																<m:munder id="id77251">
																	<m:mo id="id77252" movablelimits="false">∏</m:mo>
																	<m:mrow id="id77257">
																		<m:mn id="id77258">1</m:mn>
																		<m:mo id="id77260">≤</m:mo>
																		<m:mi id="id77263">i</m:mi>
																		<m:mo id="id77265">≤</m:mo>
																		<m:mi id="id77267">k</m:mi>
																	</m:mrow>
																</m:munder>
																<m:mfrac id="id77269">
																	<m:mrow id="id77270">
																		<m:msub id="id77272">
																			<m:mi id="id77273">c</m:mi>
																			<m:mn id="id77275">1</m:mn>
																		</m:msub>
																		<m:mo id="id77277">⁢</m:mo>
																		<m:mfenced id="id77279" open="(" close=")">
																			<m:msub id="id77284">
																				<m:mi id="id77285">L</m:mi>
																				<m:mi id="id77288">i</m:mi>
																			</m:msub>
																		</m:mfenced>
																	</m:mrow>
																	<m:mrow id="id77290">
																		<m:mn id="id77291">1</m:mn>
																		<m:mo id="id77293">-</m:mo>
																		<m:mrow id="id77295">
																			<m:mi id="id77296">exp</m:mi>
																			<m:mo id="id77298">⁡</m:mo>
																			<m:mfenced id="id77301" open="(" close=")">
																				<m:mrow id="id77306">
																					<m:mo id="id77307">-</m:mo>
																					<m:mrow id="id77309">
																						<m:msub id="id77310">
																							<m:mi id="id77311">c</m:mi>
																							<m:mn id="id77313">1</m:mn>
																						</m:msub>
																						<m:mo id="id77315">⁢</m:mo>
																						<m:mfenced id="id77318" open="(" close=")">
																							<m:msub id="id77323">
																								<m:mi id="id77324">L</m:mi>
																								<m:mi id="id77326">i</m:mi>
																							</m:msub>
																						</m:mfenced>
																					</m:mrow>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																	</m:mrow>
																</m:mfrac>
															</m:mrow>
														</m:mrow>
														<m:mo id="id77328">,</m:mo>
														<m:mrow id="id77330">
															<m:mrow id="id77331">
																<m:mrow id="id77332">
																	<m:mi id="id77333">c</m:mi>
																	<m:mo id="id77336">⁢</m:mo>
																	<m:mi id="id77338">h</m:mi>
																</m:mrow>
																<m:mo id="id77340">⁢</m:mo>
																<m:mfenced id="id77342" open="(" close=")">
																	<m:mi id="id77348">V</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id77350">=</m:mo>
															<m:mrow id="id77352">
																<m:munder id="id77353">
																	<m:mo id="id77354" movablelimits="false">∑</m:mo>
																	<m:mrow id="id77359">
																		<m:mn id="id77360">1</m:mn>
																		<m:mo id="id77362">≤</m:mo>
																		<m:mi id="id77364">i</m:mi>
																		<m:mo id="id77366">≤</m:mo>
																		<m:mi id="id77369">k</m:mi>
																	</m:mrow>
																</m:munder>
																<m:mi id="id77371">exp</m:mi>
																<m:mo id="id77373">⁡</m:mo>
																<m:mfenced id="id77375" open="(" close=")">
																	<m:mrow id="id77380">
																		<m:msub id="id77382">
																			<m:mi id="id77383">c</m:mi>
																			<m:mn id="id77385">1</m:mn>
																		</m:msub>
																		<m:mo id="id77387">⁢</m:mo>
																		<m:mfenced id="id77389" open="(" close=")">
																			<m:msub id="id77394">
																				<m:mi id="id77395">L</m:mi>
																				<m:mi id="id77398">i</m:mi>
																			</m:msub>
																		</m:mfenced>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:mo id="id77400">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id77402" encoding="MathML-Content">
													<m:apply id="id77405">
														<m:ci id="id77406"/>
														<m:apply id="id77407">
															<m:eq id="id77408"/>
															<m:apply id="id77409">
																<m:times id="id77410"/>
																<m:ci id="id77412"/>
																<m:ci id="id77413">V</m:ci>
															</m:apply>
															<m:apply id="id77415">
																<m:apply id="id77416">
																	<m:csymbol id="id77417" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id77422">∏</m:ci>
																	<m:apply id="id77424">
																		<m:ci id="id77425"/>
																		<m:cn id="id77426">1</m:cn>
																		<m:leq id="id77428"/>
																		<m:ci id="id77429">i</m:ci>
																		<m:leq id="id77431"/>
																		<m:ci id="id77432">k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id77435">
																	<m:divide id="id77436"/>
																	<m:apply id="id77437">
																		<m:times id="id77438"/>
																		<m:apply id="id77439">
																			<m:csymbol id="id77440" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id77445">c</m:ci>
																			<m:cn id="id77447">1</m:cn>
																		</m:apply>
																		<m:apply id="id77449">
																			<m:csymbol id="id77450" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id77455">L</m:ci>
																			<m:ci id="id77457">i</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id77459">
																		<m:minus id="id77460"/>
																		<m:cn id="id77461">1</m:cn>
																		<m:apply id="id77463">
																			<m:exp id="id77464"/>
																			<m:apply id="id77465">
																				<m:minus id="id77466"/>
																				<m:apply id="id77467">
																					<m:times id="id77468"/>
																					<m:apply id="id77470">
																						<m:csymbol id="id77471" cd="ambiguous">subscript</m:csymbol>
																						<m:ci id="id77475">c</m:ci>
																						<m:cn id="id77477">1</m:cn>
																					</m:apply>
																					<m:apply id="id77480">
																						<m:csymbol id="id77481" cd="ambiguous">subscript</m:csymbol>
																						<m:ci id="id77485">L</m:ci>
																						<m:ci id="id77487">i</m:ci>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id77490">
															<m:eq id="id77491"/>
															<m:apply id="id77492">
																<m:times id="id77493"/>
																<m:ci id="id77494"/>
																<m:ci id="id77495">V</m:ci>
															</m:apply>
															<m:apply id="id77497">
																<m:apply id="id77498">
																	<m:csymbol id="id77499" cd="ambiguous">subscript</m:csymbol>
																	<m:sum id="id77504"/>
																	<m:apply id="id77505">
																		<m:ci id="id77506"/>
																		<m:cn id="id77507">1</m:cn>
																		<m:leq id="id77510"/>
																		<m:ci id="id77511">i</m:ci>
																		<m:leq id="id77513"/>
																		<m:ci id="id77514">k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id77516">
																	<m:exp id="id77517"/>
																	<m:apply id="id77518">
																		<m:times id="id77519"/>
																		<m:apply id="id77520">
																			<m:csymbol id="id77521" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id77526">c</m:ci>
																			<m:cn id="id77528">1</m:cn>
																		</m:apply>
																		<m:apply id="id77530">
																			<m:csymbol id="id77531" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id77536">L</m:ci>
																			<m:ci id="id77538">i</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td id="id77541" class="eqpad"/>
								</tr>

							</table>
						</div>

						<div class="para" id="S4.p4">
							<p id="id77554" style="" class="p">We use the maps <m:math id="id77561" display="inline">
									<m:semantics id="id77564">
										<m:mrow id="id77565">
											<m:mi id="id77566">i</m:mi>
											<m:mo id="id77568">:</m:mo>
											<m:msubsup id="id77571">
												<m:mi id="id77572">μ</m:mi>
												<m:mi id="id77574">G</m:mi>
												<m:mrow id="id77576">
													<m:mo id="id77577">-</m:mo>
													<m:mn id="id77579">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id77582">(</m:mo>
											<m:mn id="id77584">0</m:mn>
											<m:mo id="id77586">)</m:mo>
											<m:mo id="id77588">/</m:mo>
											<m:mi id="id77590">T</m:mi>
											<m:mo id="id77592">↪</m:mo>
											<m:mi id="id77595">X</m:mi>
											<m:mo id="id77597">/</m:mo>
											<m:mo id="id77599">/</m:mo>
											<m:mi id="id77601">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id77603" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> and <m:math id="id77608" display="inline">
									<m:semantics id="id77611">
										<m:mrow id="id77612">
											<m:mi id="id77613">π</m:mi>
											<m:mo id="id77616">:</m:mo>
											<m:msubsup id="id77618">
												<m:mi id="id77619">μ</m:mi>
												<m:mi id="id77621">G</m:mi>
												<m:mrow id="id77623">
													<m:mo id="id77624">-</m:mo>
													<m:mn id="id77627">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id77629">(</m:mo>
											<m:mn id="id77631">0</m:mn>
											<m:mo id="id77633">)</m:mo>
											<m:mo id="id77635">/</m:mo>
											<m:mi id="id77637">T</m:mi>
											<m:mo id="id77639">→</m:mo>
											<m:mi id="id77642">X</m:mi>
											<m:mo id="id77644">/</m:mo>
											<m:mo id="id77646">/</m:mo>
											<m:mi id="id77648">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id77650" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> as defined in section <a id="id77657" href="#S1" title="1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id77655" style="" class="text tag">1</span>
								</a>, and we extend the
definition of the ‘lift’ of a cohomology class to both vector
bundles and <m:math id="id77673" display="inline">
									<m:semantics id="id77676">
										<m:mi id="id77677">K</m:mi>
										<m:annotation-xml id="id77680" encoding="MathML-Content">
											<m:ci id="id77683">K</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-theory in the obvious way. Thus we say a class
								<m:math id="id77686" display="inline">
									<m:semantics id="id77689">
										<m:mrow id="id77690">
											<m:mover id="id77692" accent="true">
												<m:mi id="id77695">a</m:mi>
												<m:mo id="id77697">~</m:mo>
											</m:mover>
											<m:mo id="id77699">∈</m:mo>
											<m:mi id="id77702">K</m:mi>
											<m:mo id="id77704">(</m:mo>
											<m:mi id="id77706">X</m:mi>
											<m:mo id="id77708">/</m:mo>
											<m:mo id="id77710">/</m:mo>
											<m:mi id="id77712">T</m:mi>
											<m:mo id="id77714">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id77716" encoding="MathML-Content">
											<m:apply id="id77720">
												<m:ci id="id77721">~</m:ci>
												<m:ci id="id77723">a</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a <span id="id77726" style="" class="text bold">lift</span> of <m:math id="id77734" display="inline">
									<m:semantics id="id77738">
										<m:mrow id="id77739">
											<m:mi id="id77740">a</m:mi>
											<m:mo id="id77742">∈</m:mo>
											<m:mi id="id77744">K</m:mi>
											<m:mo id="id77747">(</m:mo>
											<m:mi id="id77749">X</m:mi>
											<m:mo id="id77751">/</m:mo>
											<m:mo id="id77753">/</m:mo>
											<m:mi id="id77755">G</m:mi>
											<m:mo id="id77757">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id77759" encoding="MathML-Content">
											<m:ci id="id77763">a</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> if
								<m:math id="id77766" display="inline">
									<m:semantics id="id77770">
										<m:mrow id="id77771">
											<m:mrow id="id77772">
												<m:msup id="id77773">
													<m:mi id="id77774">π</m:mi>
													<m:mo id="id77776">*</m:mo>
												</m:msup>
												<m:mo id="id77778">⁢</m:mo>
												<m:mi id="id77781">a</m:mi>
											</m:mrow>
											<m:mo id="id77783">=</m:mo>
											<m:mrow id="id77785">
												<m:msup id="id77786">
													<m:mi id="id77787">i</m:mi>
													<m:mo id="id77789">*</m:mo>
												</m:msup>
												<m:mo id="id77791">⁢</m:mo>
												<m:mover id="id77794" accent="true">
													<m:mi id="id77797">a</m:mi>
													<m:mo id="id77799">~</m:mo>
												</m:mover>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id77801" encoding="MathML-Content">
											<m:apply id="id77805">
												<m:eq id="id77806"/>
												<m:apply id="id77807">
													<m:times id="id77808"/>
													<m:apply id="id77809">
														<m:csymbol id="id77810" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id77815">π</m:ci>
														<m:times id="id77817"/>
													</m:apply>
													<m:ci id="id77818">a</m:ci>
												</m:apply>
												<m:apply id="id77820">
													<m:times id="id77821"/>
													<m:apply id="id77822">
														<m:csymbol id="id77824" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id77828">i</m:ci>
														<m:times id="id77830"/>
													</m:apply>
													<m:apply id="id77831">
														<m:ci id="id77832">~</m:ci>
														<m:ci id="id77835">a</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div id="id77840" class="proof">
							<h6 id="id77844" class="title proof-title italic">Proof of theorem C.</h6>
							<div class="para" id="S4.p5">
								<p id="id77857" style="" class="p">Fix almost complex structures on <m:math id="id77864" display="inline">
										<m:semantics id="id77867">
											<m:mrow id="id77868">
												<m:mi id="id77869">X</m:mi>
												<m:mo id="id77871">/</m:mo>
												<m:mo id="id77873">/</m:mo>
												<m:mi id="id77876">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id77878" encoding="MathML-Content">
												<m:ci id="id77881">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id77884" display="inline">
										<m:semantics id="id77888">
											<m:mrow id="id77889">
												<m:mi id="id77890">X</m:mi>
												<m:mo id="id77892">/</m:mo>
												<m:mo id="id77894">/</m:mo>
												<m:mi id="id77896">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id77898" encoding="MathML-Content">
												<m:ci id="id77902">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, compatible with
their respctive symplectic forms. Throughout this proof we will let
									<m:math id="id77906" display="inline">
										<m:semantics id="id77909">
											<m:mrow id="id77910">
												<m:mi id="id77911">T</m:mi>
												<m:mo id="id77913">(</m:mo>
												<m:mi id="id77916">X</m:mi>
												<m:mo id="id77918">/</m:mo>
												<m:mo id="id77920">/</m:mo>
												<m:mi id="id77922">G</m:mi>
												<m:mo id="id77924">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id77926" encoding="MathML-Content">
												<m:ci id="id77930">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id77933" display="inline">
										<m:semantics id="id77936">
											<m:mrow id="id77937">
												<m:mi id="id77938">T</m:mi>
												<m:mo id="id77941">(</m:mo>
												<m:mi id="id77943">X</m:mi>
												<m:mo id="id77945">/</m:mo>
												<m:mo id="id77947">/</m:mo>
												<m:mi id="id77949">T</m:mi>
												<m:mo id="id77951">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id77953" encoding="MathML-Content">
												<m:ci id="id77957">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> denote the tangent bundles, thought of as
complex vector bundles given by these almost complex structures.</p>
							</div>

							<div class="para" id="S4.p6">
								<p id="id77970" style="" class="p">Define <m:math id="id77976" display="inline">
										<m:semantics id="id77980">
											<m:mrow id="id77981">
												<m:mi id="id77982">E</m:mi>
												<m:mo id="id77984">:=</m:mo>
												<m:mrow id="id77986">
													<m:msub id="id77987">
														<m:mo id="id77988">⊕</m:mo>
														<m:mrow id="id77990">
															<m:mi id="id77992">α</m:mi>
															<m:mo id="id77994">∈</m:mo>
															<m:msup id="id77996">
																<m:mi id="id77997" mathvariant="normal">Δ</m:mi>
																<m:mo id="id78002">+</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:msub id="id78004">
														<m:mi id="id78005">L</m:mi>
														<m:mi id="id78007">α</m:mi>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id78010" encoding="MathML-Content">
												<m:apply id="id78013">
													<m:ci id="id78014">:=</m:ci>
													<m:ci id="id78016">E</m:ci>
													<m:apply id="id78018">
														<m:apply id="id78020">
															<m:csymbol id="id78021" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id78025">⊕</m:ci>
															<m:apply id="id78028">
																<m:ci id="id78029">∈</m:ci>
																<m:ci id="id78031">α</m:ci>
																<m:apply id="id78034">
																	<m:csymbol id="id78035" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id78039">Δ</m:ci>
																	<m:plus id="id78042"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id78043">
															<m:csymbol id="id78044" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id78048">L</m:ci>
															<m:ci id="id78051">α</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, as in the statement of theorem C. Then we have
									<m:math id="id78055" display="inline">
										<m:semantics id="id78058">
											<m:mrow id="id78059">
												<m:msup id="id78060">
													<m:mi id="id78062">E</m:mi>
													<m:mo id="id78064">*</m:mo>
												</m:msup>
												<m:mo id="id78066">≅</m:mo>
												<m:mrow id="id78068">
													<m:msub id="id78069">
														<m:mo id="id78070">⊕</m:mo>
														<m:mrow id="id78073">
															<m:mi id="id78074">α</m:mi>
															<m:mo id="id78076">∈</m:mo>
															<m:msup id="id78079">
																<m:mi id="id78080" mathvariant="normal">Δ</m:mi>
																<m:mo id="id78084">-</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:msub id="id78086">
														<m:mi id="id78088">L</m:mi>
														<m:mi id="id78090">α</m:mi>
													</m:msub>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id78092" encoding="MathML-Content">
												<m:apply id="id78095">
													<m:approx id="id78096"/>
													<m:apply id="id78098">
														<m:csymbol id="id78099" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id78103">E</m:ci>
														<m:times id="id78105"/>
													</m:apply>
													<m:apply id="id78106">
														<m:apply id="id78108">
															<m:csymbol id="id78109" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id78113">⊕</m:ci>
															<m:apply id="id78116">
																<m:ci id="id78117">∈</m:ci>
																<m:ci id="id78119">α</m:ci>
																<m:apply id="id78122">
																	<m:csymbol id="id78123" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id78127">Δ</m:ci>
																	<m:minus id="id78130"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id78131">
															<m:csymbol id="id78132" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id78136">L</m:ci>
															<m:ci id="id78139">α</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
The main topological result, proposition <a id="id78145" href="#S1.E2" title="Proposition 1.2 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id78143" style="" class="text tag">1.2</span>
									</a>, implies that
									<m:math id="id78160" display="inline">
										<m:semantics id="id78163">
											<m:mrow id="id78164">
												<m:mo id="id78165">[</m:mo>
												<m:mi id="id78167">T</m:mi>
												<m:mo id="id78169">(</m:mo>
												<m:mi id="id78171">X</m:mi>
												<m:mo id="id78173">/</m:mo>
												<m:mo id="id78175">/</m:mo>
												<m:mi id="id78178">T</m:mi>
												<m:mo id="id78180">)</m:mo>
												<m:mo id="id78182">]</m:mo>
												<m:mo id="id78184">-</m:mo>
												<m:mo id="id78186">[</m:mo>
												<m:mi id="id78188">E</m:mi>
												<m:mo id="id78190">⊕</m:mo>
												<m:msup id="id78193">
													<m:mi id="id78194">E</m:mi>
													<m:mo id="id78196">*</m:mo>
												</m:msup>
												<m:mo id="id78198">]</m:mo>
												<m:mo id="id78200">∈</m:mo>
												<m:mi id="id78203">K</m:mi>
												<m:mo id="id78205">(</m:mo>
												<m:mi id="id78207">X</m:mi>
												<m:mo id="id78209">/</m:mo>
												<m:mo id="id78211">/</m:mo>
												<m:mi id="id78213">T</m:mi>
												<m:mo id="id78215">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id78218" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> is a lift of <m:math id="id78222" display="inline">
										<m:semantics id="id78226">
											<m:mrow id="id78227">
												<m:mo id="id78228">[</m:mo>
												<m:mi id="id78230">T</m:mi>
												<m:mo id="id78232">(</m:mo>
												<m:mi id="id78234">X</m:mi>
												<m:mo id="id78236">/</m:mo>
												<m:mo id="id78238">/</m:mo>
												<m:mi id="id78240">G</m:mi>
												<m:mo id="id78243">)</m:mo>
												<m:mo id="id78245">]</m:mo>
												<m:mo id="id78247">∈</m:mo>
												<m:mi id="id78249">K</m:mi>
												<m:mo id="id78251">(</m:mo>
												<m:mi id="id78254">X</m:mi>
												<m:mo id="id78256">/</m:mo>
												<m:mo id="id78258">/</m:mo>
												<m:mi id="id78260">G</m:mi>
												<m:mo id="id78262">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id78264" encoding="MathML-Content">
												<m:ci id="id78268">[</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Since taking characteristic classes commutes with pullback,
it follows that</p>
								<table class="equation" id="S4.Ex3">

									<tr valign="baseline" class="equation" id="S4.Ex3">
										<td id="id78288" class="eqpad"/>
										<td id="id78293" nowrap="yes" align="center" colspan="1">
											<m:math id="id78300" display="block">
												<m:semantics id="id78303">
													<m:mfrac id="id78304">
														<m:mrow id="id78305">
															<m:mrow id="id78306">
																<m:mi id="id78308">T</m:mi>
																<m:mo id="id78310">⁢</m:mo>
																<m:mi id="id78312">d</m:mi>
															</m:mrow>
															<m:mi id="id78314">T</m:mi>
															<m:mo id="id78316">(</m:mo>
															<m:mi id="id78318">X</m:mi>
															<m:mo id="id78321">/</m:mo>
															<m:mo id="id78323">/</m:mo>
															<m:mi id="id78325">T</m:mi>
															<m:mo id="id78327">)</m:mo>
														</m:mrow>
														<m:mrow id="id78329">
															<m:mrow id="id78330">
																<m:mrow id="id78331">
																	<m:mi id="id78332">T</m:mi>
																	<m:mo id="id78334">⁢</m:mo>
																	<m:mi id="id78337">d</m:mi>
																</m:mrow>
																<m:mo id="id78339">⁢</m:mo>
																<m:mfenced id="id78341" open="(" close=")">
																	<m:mi id="id78346">E</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id78349">⌣</m:mo>
															<m:mrow id="id78351">
																<m:mrow id="id78352">
																	<m:mi id="id78353">T</m:mi>
																	<m:mo id="id78355">⁢</m:mo>
																	<m:mi id="id78358">d</m:mi>
																</m:mrow>
																<m:mo id="id78360">⁢</m:mo>
																<m:mfenced id="id78362" open="(" close=")">
																	<m:msup id="id78367">
																		<m:mi id="id78368">E</m:mi>
																		<m:mo id="id78370">*</m:mo>
																	</m:msup>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mfrac>
													<m:annotation-xml id="id78373" encoding="MathML-Content">
														<m:apply id="id78376">
															<m:divide id="id78377"/>
															<m:mtext id="id78378">⁢TdT(X//T)</m:mtext>
															<m:apply id="id78380">
																<m:ci id="id78382">⌣</m:ci>
																<m:apply id="id78384">
																	<m:times id="id78385"/>
																	<m:ci id="id78386"/>
																	<m:ci id="id78387">E</m:ci>
																</m:apply>
																<m:apply id="id78389">
																	<m:times id="id78390"/>
																	<m:ci id="id78391"/>
																	<m:apply id="id78392">
																		<m:csymbol id="id78394" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id78398">E</m:ci>
																		<m:times id="id78400"/>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id78402" class="eqpad"/>
									</tr>

								</table>
								<p id="id78409" style="" class="p">is a lift of <m:math id="id78415" display="inline">
										<m:semantics id="id78419">
											<m:mrow id="id78420">
												<m:mrow id="id78421">
													<m:mi id="id78422">T</m:mi>
													<m:mo id="id78424">⁢</m:mo>
													<m:mi id="id78426">d</m:mi>
												</m:mrow>
												<m:mi id="id78429">T</m:mi>
												<m:mo id="id78431">(</m:mo>
												<m:mi id="id78433">X</m:mi>
												<m:mo id="id78435">/</m:mo>
												<m:mo id="id78437">/</m:mo>
												<m:mi id="id78439">G</m:mi>
												<m:mo id="id78441">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id78444" encoding="MathML-Content">
												<m:ci id="id78447"/>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S4.p7">
								<p id="id78458" style="" class="p">Define the class <m:math id="id78464" display="inline">
										<m:semantics id="id78468">
											<m:mrow id="id78469">
												<m:mi id="id78470">b</m:mi>
												<m:mo id="id78472">∈</m:mo>
												<m:msup id="id78474">
													<m:mi id="id78475">H</m:mi>
													<m:mo id="id78477">*</m:mo>
												</m:msup>
												<m:mo id="id78480">(</m:mo>
												<m:mi id="id78482">X</m:mi>
												<m:mo id="id78484">/</m:mo>
												<m:mo id="id78486">/</m:mo>
												<m:mi id="id78488">T</m:mi>
												<m:mo id="id78490">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id78492" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> by <m:math id="id78497" display="inline">
										<m:semantics id="id78500">
											<m:mrow id="id78501">
												<m:mi id="id78502">b</m:mi>
												<m:mo id="id78505">:=</m:mo>
												<m:mrow id="id78507">
													<m:mrow id="id78508">
														<m:mi id="id78509">T</m:mi>
														<m:mo id="id78511">⁢</m:mo>
														<m:mi id="id78513">d</m:mi>
													</m:mrow>
													<m:mo id="id78516">⁢</m:mo>
													<m:mfenced id="id78518" open="(" close=")">
														<m:mi id="id78523">E</m:mi>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id78525" encoding="MathML-Content">
												<m:apply id="id78528">
													<m:ci id="id78530">:=</m:ci>
													<m:ci id="id78532">b</m:ci>
													<m:apply id="id78534">
														<m:times id="id78535"/>
														<m:ci id="id78536"/>
														<m:ci id="id78537">E</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Then
									<m:math id="id78540" display="inline">
										<m:semantics id="id78544">
											<m:mrow id="id78545">
												<m:mrow id="id78546">
													<m:msup id="id78547">
														<m:mi id="id78548">i</m:mi>
														<m:mo id="id78550">*</m:mo>
													</m:msup>
													<m:mo id="id78552">⁢</m:mo>
													<m:mi id="id78555">b</m:mi>
												</m:mrow>
												<m:mo id="id78557">=</m:mo>
												<m:mrow id="id78559">
													<m:mrow id="id78560">
														<m:mi id="id78561">T</m:mi>
														<m:mo id="id78563">⁢</m:mo>
														<m:mi id="id78566">d</m:mi>
													</m:mrow>
													<m:mo id="id78568">⁢</m:mo>
													<m:mfenced id="id78570" open="(" close=")">
														<m:mrow id="id78575">
															<m:mrow id="id78576">
																<m:mi id="id78577">v</m:mi>
																<m:mo id="id78580">⁢</m:mo>
																<m:mi id="id78582">e</m:mi>
																<m:mo id="id78584">⁢</m:mo>
																<m:mi id="id78586">r</m:mi>
																<m:mo id="id78589">⁢</m:mo>
																<m:mi id="id78591">t</m:mi>
															</m:mrow>
															<m:mo id="id78593">⁢</m:mo>
															<m:mfenced id="id78596" open="(" close=")">
																<m:mi id="id78601">π</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id78603" encoding="MathML-Content">
												<m:apply id="id78606">
													<m:eq id="id78607"/>
													<m:apply id="id78608">
														<m:times id="id78610"/>
														<m:apply id="id78611">
															<m:csymbol id="id78612" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id78616">i</m:ci>
															<m:times id="id78618"/>
														</m:apply>
														<m:ci id="id78620">b</m:ci>
													</m:apply>
													<m:apply id="id78622">
														<m:times id="id78623"/>
														<m:ci id="id78624"/>
														<m:apply id="id78625">
															<m:times id="id78626"/>
															<m:ci id="id78627"/>
															<m:ci id="id78628">π</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Now <m:math id="id78632" display="inline">
										<m:semantics id="id78635">
											<m:mi id="id78636">π</m:mi>
											<m:annotation-xml id="id78639" encoding="MathML-Content">
												<m:ci id="id78642">π</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is the projection of a fibre
bundle, with fibres <m:math id="id78646" display="inline">
										<m:semantics id="id78649">
											<m:mrow id="id78650">
												<m:mi id="id78651">G</m:mi>
												<m:mo id="id78654">/</m:mo>
												<m:mi id="id78656">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id78658" encoding="MathML-Content">
												<m:apply id="id78661">
													<m:divide id="id78662"/>
													<m:ci id="id78663">G</m:ci>
													<m:ci id="id78665">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which arises as the global quotient of a
principal <m:math id="id78669" display="inline">
										<m:semantics id="id78672">
											<m:mi id="id78674">G</m:mi>
											<m:annotation-xml id="id78676" encoding="MathML-Content">
												<m:ci id="id78679">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> bundle by the maximal torus <m:math id="id78683" display="inline">
										<m:semantics id="id78686">
											<m:mi id="id78687">T</m:mi>
											<m:annotation-xml id="id78689" encoding="MathML-Content">
												<m:ci id="id78692">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. A result of Borel and
Hirzebruch asserts that in this case <m:math id="id78696" display="inline">
										<m:semantics id="id78700">
											<m:mrow id="id78701">
												<m:mrow id="id78702">
													<m:msub id="id78703">
														<m:mi id="id78704">π</m:mi>
														<m:mo id="id78706">*</m:mo>
													</m:msub>
													<m:mo id="id78708">⁢</m:mo>
													<m:mrow id="id78711">
														<m:mi id="id78712">T</m:mi>
														<m:mo id="id78714">⁢</m:mo>
														<m:mi id="id78716">d</m:mi>
													</m:mrow>
													<m:mo id="id78719">⁢</m:mo>
													<m:mfenced id="id78721" open="(" close=")">
														<m:mrow id="id78726">
															<m:mrow id="id78727">
																<m:mi id="id78728">v</m:mi>
																<m:mo id="id78730">⁢</m:mo>
																<m:mi id="id78733">e</m:mi>
																<m:mo id="id78735">⁢</m:mo>
																<m:mi id="id78737">r</m:mi>
																<m:mo id="id78739">⁢</m:mo>
																<m:mi id="id78742">t</m:mi>
															</m:mrow>
															<m:mo id="id78744">⁢</m:mo>
															<m:mfenced id="id78746" open="(" close=")">
																<m:mi id="id78751">π</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id78754">=</m:mo>
												<m:mn id="id78756">1</m:mn>
											</m:mrow>
											<m:annotation-xml id="id78758" encoding="MathML-Content">
												<m:apply id="id78761">
													<m:eq id="id78762"/>
													<m:apply id="id78764">
														<m:times id="id78765"/>
														<m:apply id="id78766">
															<m:csymbol id="id78767" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id78771">π</m:ci>
															<m:times id="id78774"/>
														</m:apply>
														<m:ci id="id78775"/>
														<m:apply id="id78776">
															<m:times id="id78777"/>
															<m:ci id="id78778"/>
															<m:ci id="id78779">π</m:ci>
														</m:apply>
													</m:apply>
													<m:cn id="id78782">1</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
									<cite id="id78785" class="cite">[<a id="id78789" href="#bib.bib3" title="" class="ref">3</a>, sections 7.4 and 22.3]</cite>.
								</p>
							</div>

							<div class="para" id="S4.p8">
								<p id="id78809" style="" class="p">Applying the Atiyah-Singer index theorem, and arguing as in the proof
of the integration formula,</p>
								<table class="equation" id="S4.Ex4">

									<tr valign="baseline" class="equation" id="S4.Ex4">
										<td id="id78833" class="eqpad"/>
										<td id="id78837" nowrap="yes" align="center" colspan="1">
											<m:math id="id78844" display="block">
												<m:semantics id="id78848">
													<m:mrow id="id78849">
														<m:msup id="id78850">
															<m:mrow id="id78851">
																<m:mi id="id78852">i</m:mi>
																<m:mo id="id78854">⁢</m:mo>
																<m:mi id="id78856">n</m:mi>
																<m:mo id="id78859">⁢</m:mo>
																<m:mi id="id78861">d</m:mi>
																<m:mo id="id78863">⁢</m:mo>
																<m:mi id="id78866">e</m:mi>
																<m:mo id="id78868">⁢</m:mo>
																<m:mi id="id78870">x</m:mi>
															</m:mrow>
															<m:mrow id="id78872">
																<m:mi id="id78873">X</m:mi>
																<m:mo id="id78875">⁣</m:mo>
																<m:mo id="id78878">/</m:mo>
																<m:mo id="id78880">⁣</m:mo>
																<m:mrow id="id78882">
																	<m:none id="id78883"/>
																	<m:mo id="id78884">/</m:mo>
																	<m:mi id="id78887">G</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msup>
														<m:msub id="id78889">
															<m:mi id="id78890">D</m:mi>
															<m:mi id="id78892">V</m:mi>
														</m:msub>
														<m:mo id="id78894">=</m:mo>
														<m:msub id="id78896">
															<m:mo id="id78897">∫</m:mo>
															<m:mrow id="id78900">
																<m:mi id="id78901">X</m:mi>
																<m:mo id="id78903">⁣</m:mo>
																<m:mo id="id78905">/</m:mo>
																<m:mo id="id78907">⁣</m:mo>
																<m:mrow id="id78910">
																	<m:none id="id78911"/>
																	<m:mo id="id78912">/</m:mo>
																	<m:mi id="id78914">G</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id78916">
															<m:mi id="id78917">c</m:mi>
															<m:mo id="id78919">⁢</m:mo>
															<m:mi id="id78922">h</m:mi>
														</m:mrow>
														<m:mo id="id78924">(</m:mo>
														<m:mi id="id78926">V</m:mi>
														<m:mo id="id78928">)</m:mo>
														<m:mo id="id78930">⌣</m:mo>
														<m:mrow id="id78933">
															<m:mi id="id78934">T</m:mi>
															<m:mo id="id78936">⁢</m:mo>
															<m:mi id="id78938">d</m:mi>
														</m:mrow>
														<m:mi id="id78940">T</m:mi>
														<m:mo id="id78943">(</m:mo>
														<m:mi id="id78945">X</m:mi>
														<m:mo id="id78947">/</m:mo>
														<m:mo id="id78949">/</m:mo>
														<m:mi id="id78951">G</m:mi>
														<m:mo id="id78953">)</m:mo>
														<m:mo id="id78955">=</m:mo>
														<m:msub id="id78958">
															<m:mo id="id78959">∫</m:mo>
															<m:mrow id="id78961">
																<m:mi id="id78962">X</m:mi>
																<m:mo id="id78964">⁣</m:mo>
																<m:mo id="id78967">/</m:mo>
																<m:mo id="id78969">⁣</m:mo>
																<m:mrow id="id78971">
																	<m:none id="id78972"/>
																	<m:mo id="id78973">/</m:mo>
																	<m:mi id="id78975">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id78978">
															<m:mi id="id78979">c</m:mi>
															<m:mo id="id78981">⁢</m:mo>
															<m:mi id="id78983">h</m:mi>
														</m:mrow>
														<m:mo id="id78985">(</m:mo>
														<m:mover id="id78987" accent="true">
															<m:mi id="id78991">V</m:mi>
															<m:mo id="id78993">~</m:mo>
														</m:mover>
														<m:mo id="id78995">)</m:mo>
														<m:mo id="id78997">⌣</m:mo>
														<m:mfrac id="id79000">
															<m:mrow id="id79001">
																<m:mrow id="id79002">
																	<m:mi id="id79003">T</m:mi>
																	<m:mo id="id79005">⁢</m:mo>
																	<m:mi id="id79007">d</m:mi>
																</m:mrow>
																<m:mi id="id79009">T</m:mi>
																<m:mo id="id79012">(</m:mo>
																<m:mi id="id79014">X</m:mi>
																<m:mo id="id79016">/</m:mo>
																<m:mo id="id79018">/</m:mo>
																<m:mi id="id79020">T</m:mi>
																<m:mo id="id79022">)</m:mo>
															</m:mrow>
															<m:mrow id="id79024">
																<m:mrow id="id79025">
																	<m:mrow id="id79026">
																		<m:mi id="id79028">T</m:mi>
																		<m:mo id="id79030">⁢</m:mo>
																		<m:mi id="id79032">d</m:mi>
																	</m:mrow>
																	<m:mo id="id79034">⁢</m:mo>
																	<m:mfenced id="id79037" open="(" close=")">
																		<m:mi id="id79042">E</m:mi>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id79044">⌣</m:mo>
																<m:mrow id="id79046">
																	<m:mrow id="id79047">
																		<m:mi id="id79048">T</m:mi>
																		<m:mo id="id79050">⁢</m:mo>
																		<m:mi id="id79053">d</m:mi>
																	</m:mrow>
																	<m:mo id="id79055">⁢</m:mo>
																	<m:mfenced id="id79057" open="(" close=")">
																		<m:msup id="id79062">
																			<m:mi id="id79064">E</m:mi>
																			<m:mo id="id79066">*</m:mo>
																		</m:msup>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
														</m:mfrac>
														<m:mo id="id79068">⌣</m:mo>
														<m:mi id="id79070">b</m:mi>
														<m:mo id="id79072">⌣</m:mo>
														<m:munder id="id79075">
															<m:mo id="id79076" movablelimits="false">∏</m:mo>
															<m:mrow id="id79080">
																<m:mi id="id79082">α</m:mi>
																<m:mo id="id79084">∈</m:mo>
																<m:msup id="id79086">
																	<m:mi id="id79087" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id79092">-</m:mo>
																</m:msup>
															</m:mrow>
														</m:munder>
														<m:mi id="id79094">e</m:mi>
														<m:mo id="id79096">(</m:mo>
														<m:mi id="id79098">α</m:mi>
														<m:mo id="id79101">)</m:mo>
														<m:mo id="id79103">=</m:mo>
														<m:msub id="id79105">
															<m:mo id="id79106">∫</m:mo>
															<m:mrow id="id79109">
																<m:mi id="id79110">X</m:mi>
																<m:mo id="id79112">⁣</m:mo>
																<m:mo id="id79114">/</m:mo>
																<m:mo id="id79116">⁣</m:mo>
																<m:mrow id="id79119">
																	<m:none id="id79120"/>
																	<m:mo id="id79121">/</m:mo>
																	<m:mi id="id79123">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id79125">
															<m:mi id="id79126">c</m:mi>
															<m:mo id="id79128">⁢</m:mo>
															<m:mi id="id79131">h</m:mi>
														</m:mrow>
														<m:mo id="id79133">(</m:mo>
														<m:mover id="id79135" accent="true">
															<m:mi id="id79138">V</m:mi>
															<m:mo id="id79140">~</m:mo>
														</m:mover>
														<m:mo id="id79143">)</m:mo>
														<m:mo id="id79145">⌣</m:mo>
														<m:mfrac id="id79147">
															<m:mrow id="id79148">
																<m:mrow id="id79149">
																	<m:mi id="id79150">T</m:mi>
																	<m:mo id="id79152">⁢</m:mo>
																	<m:mi id="id79155">d</m:mi>
																</m:mrow>
																<m:mi id="id79157">T</m:mi>
																<m:mo id="id79159">(</m:mo>
																<m:mi id="id79161">X</m:mi>
																<m:mo id="id79163">/</m:mo>
																<m:mo id="id79166">/</m:mo>
																<m:mi id="id79168">T</m:mi>
																<m:mo id="id79170">)</m:mo>
															</m:mrow>
															<m:mrow id="id79172">
																<m:mrow id="id79173">
																	<m:mi id="id79174">T</m:mi>
																	<m:mo id="id79176">⁢</m:mo>
																	<m:mi id="id79179">d</m:mi>
																</m:mrow>
																<m:mo id="id79181">⁢</m:mo>
																<m:mfenced id="id79183" open="(" close=")">
																	<m:msup id="id79188">
																		<m:mi id="id79189">E</m:mi>
																		<m:mo id="id79191">*</m:mo>
																	</m:msup>
																</m:mfenced>
															</m:mrow>
														</m:mfrac>
														<m:mo id="id79194">⌣</m:mo>
														<m:munder id="id79196">
															<m:mo id="id79197" movablelimits="false">∏</m:mo>
															<m:mrow id="id79202">
																<m:mi id="id79203">α</m:mi>
																<m:mo id="id79205">∈</m:mo>
																<m:msup id="id79208">
																	<m:mi id="id79209" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id79213">-</m:mo>
																</m:msup>
															</m:mrow>
														</m:munder>
														<m:mi id="id79215">e</m:mi>
														<m:mo id="id79218">(</m:mo>
														<m:mi id="id79220">α</m:mi>
														<m:mo id="id79222">)</m:mo>
														<m:mo id="id79224">=</m:mo>
														<m:msub id="id79226">
															<m:mo id="id79227">∫</m:mo>
															<m:mrow id="id79230">
																<m:mi id="id79231">X</m:mi>
																<m:mo id="id79233">⁣</m:mo>
																<m:mo id="id79235">/</m:mo>
																<m:mo id="id79238">⁣</m:mo>
																<m:mrow id="id79240">
																	<m:none id="id79241"/>
																	<m:mo id="id79242">/</m:mo>
																	<m:mi id="id79244">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id79246">
															<m:mi id="id79247">c</m:mi>
															<m:mo id="id79250">⁢</m:mo>
															<m:mi id="id79252">h</m:mi>
														</m:mrow>
														<m:mo id="id79254">(</m:mo>
														<m:mover id="id79256" accent="true">
															<m:mi id="id79260">V</m:mi>
															<m:mo id="id79262">~</m:mo>
														</m:mover>
														<m:mo id="id79264">)</m:mo>
														<m:mo id="id79266">⌣</m:mo>
														<m:mrow id="id79268">
															<m:mi id="id79269">T</m:mi>
															<m:mo id="id79272">⁢</m:mo>
															<m:mi id="id79274">d</m:mi>
														</m:mrow>
														<m:mi id="id79276">T</m:mi>
														<m:mo id="id79278">(</m:mo>
														<m:mi id="id79280">X</m:mi>
														<m:mo id="id79282">/</m:mo>
														<m:mo id="id79285">/</m:mo>
														<m:mi id="id79287">T</m:mi>
														<m:mo id="id79289">)</m:mo>
														<m:mo id="id79291">⌣</m:mo>
														<m:munder id="id79293">
															<m:mo id="id79294" movablelimits="false">∏</m:mo>
															<m:mrow id="id79299">
																<m:mi id="id79300">α</m:mi>
																<m:mo id="id79303">∈</m:mo>
																<m:msup id="id79305">
																	<m:mi id="id79306" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id79311">-</m:mo>
																</m:msup>
															</m:mrow>
														</m:munder>
														<m:mo id="id79313">(</m:mo>
														<m:mn id="id79315">1</m:mn>
														<m:mo id="id79317">-</m:mo>
														<m:mi id="id79319">exp</m:mi>
														<m:mo id="id79321">(</m:mo>
														<m:mo id="id79324">-</m:mo>
														<m:mi id="id79326">e</m:mi>
														<m:mo id="id79328">(</m:mo>
														<m:mi id="id79330">α</m:mi>
														<m:mo id="id79332">)</m:mo>
														<m:mo id="id79334">)</m:mo>
														<m:mo id="id79337">)</m:mo>
														<m:mo id="id79339">=</m:mo>
														<m:msub id="id79341">
															<m:mo id="id79342">∫</m:mo>
															<m:mrow id="id79344">
																<m:mi id="id79345">X</m:mi>
																<m:mo id="id79348">⁣</m:mo>
																<m:mo id="id79350">/</m:mo>
																<m:mo id="id79352">⁣</m:mo>
																<m:mrow id="id79354">
																	<m:none id="id79356"/>
																	<m:mo id="id79357">/</m:mo>
																	<m:mi id="id79359">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id79361">
															<m:mi id="id79362">c</m:mi>
															<m:mo id="id79364">⁢</m:mo>
															<m:mi id="id79366">h</m:mi>
														</m:mrow>
														<m:mo id="id79369">(</m:mo>
														<m:mover id="id79371" accent="true">
															<m:mi id="id79374">V</m:mi>
															<m:mo id="id79376">~</m:mo>
														</m:mover>
														<m:mo id="id79378">)</m:mo>
														<m:mo id="id79380">⌣</m:mo>
														<m:mrow id="id79383">
															<m:mi id="id79384">T</m:mi>
															<m:mo id="id79386">⁢</m:mo>
															<m:mi id="id79388">d</m:mi>
														</m:mrow>
														<m:mi id="id79391">T</m:mi>
														<m:mo id="id79393">(</m:mo>
														<m:mi id="id79395">X</m:mi>
														<m:mo id="id79397">/</m:mo>
														<m:mo id="id79399">/</m:mo>
														<m:mi id="id79401">T</m:mi>
														<m:mo id="id79403">)</m:mo>
														<m:mo id="id79406">⌣</m:mo>
														<m:munder id="id79408">
															<m:mo id="id79409" movablelimits="false">∏</m:mo>
															<m:mrow id="id79414">
																<m:mi id="id79415">α</m:mi>
																<m:mo id="id79417">∈</m:mo>
																<m:msup id="id79420">
																	<m:mi id="id79421" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id79425">+</m:mo>
																</m:msup>
															</m:mrow>
														</m:munder>
														<m:mo id="id79427">(</m:mo>
														<m:mn id="id79430">1</m:mn>
														<m:mo id="id79432">-</m:mo>
														<m:mi id="id79434">exp</m:mi>
														<m:mo id="id79436">(</m:mo>
														<m:mi id="id79438">e</m:mi>
														<m:mo id="id79440">(</m:mo>
														<m:mi id="id79442">α</m:mi>
														<m:mo id="id79445">)</m:mo>
														<m:mo id="id79447">)</m:mo>
														<m:mo id="id79449">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id79451" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id79456" class="eqpad"/>
									</tr>

								</table>
								<p id="id79462" style="" class="p">But <m:math id="id79468" display="inline">
										<m:semantics id="id79472">
											<m:mrow id="id79473">
												<m:mrow id="id79474">
													<m:mfenced id="id79475" open="[" close="]">
														<m:mrow id="id79480">
															<m:msup id="id79481">
																<m:mi id="id79482" mathvariant="normal">Λ</m:mi>
																<m:mtext id="id79487">even</m:mtext>
															</m:msup>
															<m:mo id="id79489">⁢</m:mo>
															<m:mi id="id79492">E</m:mi>
														</m:mrow>
													</m:mfenced>
													<m:mo id="id79494">-</m:mo>
													<m:mfenced id="id79496" open="[" close="]">
														<m:mrow id="id79501">
															<m:msup id="id79502">
																<m:mi id="id79503" mathvariant="normal">Λ</m:mi>
																<m:mtext id="id79508">odd</m:mtext>
															</m:msup>
															<m:mo id="id79510">⁢</m:mo>
															<m:mi id="id79512">E</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id79514">=</m:mo>
												<m:mrow id="id79517">
													<m:msubsup id="id79518">
														<m:mo id="id79519">∑</m:mo>
														<m:mrow id="id79521">
															<m:mi id="id79522">i</m:mi>
															<m:mo id="id79524">=</m:mo>
															<m:mn id="id79526">0</m:mn>
														</m:mrow>
														<m:mrow id="id79529">
															<m:mrow id="id79530">
																<m:mi id="id79531">r</m:mi>
																<m:mo id="id79533">⁢</m:mo>
																<m:mi id="id79535">k</m:mi>
															</m:mrow>
															<m:mo id="id79537">⁢</m:mo>
															<m:mi id="id79540">E</m:mi>
														</m:mrow>
													</m:msubsup>
													<m:msup id="id79542">
														<m:mfenced id="id79543" open="(" close=")">
															<m:mrow id="id79548">
																<m:mo id="id79549">-</m:mo>
																<m:mn id="id79551">1</m:mn>
															</m:mrow>
														</m:mfenced>
														<m:mi id="id79553">i</m:mi>
													</m:msup>
													<m:mo id="id79556">⁢</m:mo>
													<m:mfenced id="id79558" open="[" close="]">
														<m:mrow id="id79563">
															<m:msup id="id79564">
																<m:mi id="id79565" mathvariant="normal">Λ</m:mi>
																<m:mi id="id79570">i</m:mi>
															</m:msup>
															<m:mo id="id79572">⁢</m:mo>
															<m:mi id="id79574">E</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id79576">=</m:mo>
												<m:mrow id="id79579">
													<m:msub id="id79580">
														<m:mo id="id79581">∏</m:mo>
														<m:mrow id="id79583">
															<m:mi id="id79584">α</m:mi>
															<m:mo id="id79587">∈</m:mo>
															<m:msup id="id79589">
																<m:mi id="id79590" mathvariant="normal">Δ</m:mi>
																<m:mo id="id79595">+</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:mfenced id="id79597" open="(" close=")">
														<m:mrow id="id79602">
															<m:mfenced id="id79603" open="[" close="]">
																<m:munder id="id79608" accent="true">
																	<m:mi id="id79611" mathvariant="double-struck">C</m:mi>
																	<m:mo id="id79616">¯</m:mo>
																</m:munder>
															</m:mfenced>
															<m:mo id="id79618">-</m:mo>
															<m:mfenced id="id79620" open="[" close="]">
																<m:msub id="id79625">
																	<m:mi id="id79626">L</m:mi>
																	<m:mi id="id79629">α</m:mi>
																</m:msub>
															</m:mfenced>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id79631" encoding="MathML-Content">
												<m:apply id="id79634">
													<m:ci id="id79635"/>
													<m:apply id="id79636">
														<m:minus id="id79638"/>
														<m:apply id="id79639">
															<m:times id="id79640"/>
															<m:apply id="id79641">
																<m:csymbol id="id79642" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id79646">Λ</m:ci>
																<m:mtext id="id79649">even</m:mtext>
															</m:apply>
															<m:ci id="id79651">E</m:ci>
														</m:apply>
														<m:apply id="id79653">
															<m:times id="id79654"/>
															<m:apply id="id79656">
																<m:csymbol id="id79657" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id79661">Λ</m:ci>
																<m:mtext id="id79664">odd</m:mtext>
															</m:apply>
															<m:ci id="id79666">E</m:ci>
														</m:apply>
													</m:apply>
													<m:eq id="id79668"/>
													<m:apply id="id79669">
														<m:apply id="id79670">
															<m:csymbol id="id79671" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id79676">
																<m:csymbol id="id79677" cd="ambiguous">subscript</m:csymbol>
																<m:sum id="id79682"/>
																<m:apply id="id79683">
																	<m:eq id="id79684"/>
																	<m:ci id="id79685">i</m:ci>
																	<m:cn id="id79687">0</m:cn>
																</m:apply>
															</m:apply>
															<m:apply id="id79689">
																<m:times id="id79690"/>
																<m:ci id="id79691"/>
																<m:ci id="id79692">E</m:ci>
															</m:apply>
														</m:apply>
														<m:apply id="id79694">
															<m:times id="id79695"/>
															<m:apply id="id79696">
																<m:csymbol id="id79698" cd="ambiguous">superscript</m:csymbol>
																<m:apply id="id79702">
																	<m:minus id="id79703"/>
																	<m:cn id="id79704">1</m:cn>
																</m:apply>
																<m:ci id="id79706">i</m:ci>
															</m:apply>
															<m:apply id="id79709">
																<m:times id="id79710"/>
																<m:apply id="id79711">
																	<m:csymbol id="id79712" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id79716">Λ</m:ci>
																	<m:ci id="id79719">i</m:ci>
																</m:apply>
																<m:ci id="id79721">E</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:eq id="id79723"/>
													<m:apply id="id79724">
														<m:apply id="id79725">
															<m:csymbol id="id79726" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id79731">∏</m:ci>
															<m:apply id="id79733">
																<m:ci id="id79734">∈</m:ci>
																<m:ci id="id79737">α</m:ci>
																<m:apply id="id79739">
																	<m:csymbol id="id79740" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id79745">Δ</m:ci>
																	<m:plus id="id79747"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id79748">
															<m:minus id="id79750"/>
															<m:apply id="id79751">
																<m:ci id="id79752">¯</m:ci>
																<m:ci id="id79754">C</m:ci>
															</m:apply>
															<m:apply id="id79756">
																<m:csymbol id="id79757" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id79762">L</m:ci>
																<m:ci id="id79764">α</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> hence,
applying the Chern character,
									<m:math id="id79768" display="inline">
										<m:semantics id="id79771">
											<m:mrow id="id79772">
												<m:mrow id="id79773">
													<m:mrow id="id79774">
														<m:mrow id="id79776">
															<m:mi id="id79777">c</m:mi>
															<m:mo id="id79779">⁢</m:mo>
															<m:mi id="id79781">h</m:mi>
														</m:mrow>
														<m:mo id="id79783">⁢</m:mo>
														<m:mfenced id="id79786" open="(" close=")">
															<m:mrow id="id79791">
																<m:msup id="id79792">
																	<m:mi id="id79793" mathvariant="normal">Λ</m:mi>
																	<m:mtext id="id79798">even</m:mtext>
																</m:msup>
																<m:mo id="id79800">⁢</m:mo>
																<m:mi id="id79802">E</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id79804">-</m:mo>
													<m:mrow id="id79807">
														<m:mrow id="id79808">
															<m:mi id="id79809">c</m:mi>
															<m:mo id="id79811">⁢</m:mo>
															<m:mi id="id79813">h</m:mi>
														</m:mrow>
														<m:mo id="id79815">⁢</m:mo>
														<m:mfenced id="id79818" open="(" close=")">
															<m:mrow id="id79823">
																<m:msup id="id79824">
																	<m:mi id="id79825" mathvariant="normal">Λ</m:mi>
																	<m:mtext id="id79830">odd</m:mtext>
																</m:msup>
																<m:mo id="id79832">⁢</m:mo>
																<m:mi id="id79834">E</m:mi>
															</m:mrow>
														</m:mfenced>
													</m:mrow>
												</m:mrow>
												<m:mo id="id79836">=</m:mo>
												<m:mrow id="id79838">
													<m:msub id="id79840">
														<m:mo id="id79841">∏</m:mo>
														<m:mrow id="id79843">
															<m:mi id="id79844">α</m:mi>
															<m:mo id="id79846">∈</m:mo>
															<m:msup id="id79849">
																<m:mi id="id79850" mathvariant="normal">Δ</m:mi>
																<m:mo id="id79855">+</m:mo>
															</m:msup>
														</m:mrow>
													</m:msub>
													<m:mfenced id="id79857" open="(" close=")">
														<m:mrow id="id79862">
															<m:mn id="id79863">1</m:mn>
															<m:mo id="id79865">-</m:mo>
															<m:mrow id="id79867">
																<m:mi id="id79868">exp</m:mi>
																<m:mo id="id79870">⁡</m:mo>
																<m:mfenced id="id79873" open="(" close=")">
																	<m:mrow id="id79878">
																		<m:mi id="id79879">e</m:mi>
																		<m:mo id="id79881">⁢</m:mo>
																		<m:mfenced id="id79883" open="(" close=")">
																			<m:mi id="id79888">α</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id79891" encoding="MathML-Content">
												<m:apply id="id79894">
													<m:eq id="id79895"/>
													<m:apply id="id79896">
														<m:minus id="id79897"/>
														<m:apply id="id79898">
															<m:times id="id79900"/>
															<m:ci id="id79901"/>
															<m:apply id="id79902">
																<m:times id="id79903"/>
																<m:apply id="id79904">
																	<m:csymbol id="id79905" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id79910">Λ</m:ci>
																	<m:mtext id="id79912">even</m:mtext>
																</m:apply>
																<m:ci id="id79914">E</m:ci>
															</m:apply>
														</m:apply>
														<m:apply id="id79916">
															<m:times id="id79918"/>
															<m:ci id="id79919"/>
															<m:apply id="id79920">
																<m:times id="id79921"/>
																<m:apply id="id79922">
																	<m:csymbol id="id79923" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id79928">Λ</m:ci>
																	<m:mtext id="id79930">odd</m:mtext>
																</m:apply>
																<m:ci id="id79932">E</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply id="id79934">
														<m:apply id="id79935">
															<m:csymbol id="id79936" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id79941">∏</m:ci>
															<m:apply id="id79943">
																<m:ci id="id79944">∈</m:ci>
																<m:ci id="id79947">α</m:ci>
																<m:apply id="id79949">
																	<m:csymbol id="id79950" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id79955">Δ</m:ci>
																	<m:plus id="id79957"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id79958">
															<m:minus id="id79960"/>
															<m:cn id="id79961">1</m:cn>
															<m:apply id="id79963">
																<m:exp id="id79964"/>
																<m:apply id="id79965">
																	<m:times id="id79966"/>
																	<m:ci id="id79967">e</m:ci>
																	<m:ci id="id79969">α</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
								</p>
							</div>

							<div class="para" id="S4.p9">
								<p id="id79981" style="" class="p">Combining these formulæ, using additive and multiplicative properties of the
Chern character, gives</p>
								<table class="equation" id="S4.Ex5">

									<tr valign="baseline" class="equation" id="S4.Ex5">
										<td id="id80006" class="eqpad"/>
										<td id="id80010" nowrap="yes" align="center" colspan="1">
											<m:math id="id80018" display="block">
												<m:semantics id="id80021">
													<m:mrow id="id80022">
														<m:msup id="id80023">
															<m:mrow id="id80024">
																<m:mi id="id80025">i</m:mi>
																<m:mo id="id80028">⁢</m:mo>
																<m:mi id="id80030">n</m:mi>
																<m:mo id="id80032">⁢</m:mo>
																<m:mi id="id80034">d</m:mi>
																<m:mo id="id80037">⁢</m:mo>
																<m:mi id="id80039">e</m:mi>
																<m:mo id="id80041">⁢</m:mo>
																<m:mi id="id80044">x</m:mi>
															</m:mrow>
															<m:mrow id="id80046">
																<m:mi id="id80047">X</m:mi>
																<m:mo id="id80049">⁣</m:mo>
																<m:mo id="id80051">/</m:mo>
																<m:mo id="id80053">⁣</m:mo>
																<m:mrow id="id80056">
																	<m:none id="id80057"/>
																	<m:mo id="id80058">/</m:mo>
																	<m:mi id="id80060">G</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msup>
														<m:msub id="id80062">
															<m:mi id="id80063">D</m:mi>
															<m:mi id="id80065">V</m:mi>
														</m:msub>
														<m:mo id="id80068">=</m:mo>
														<m:msub id="id80070">
															<m:mo id="id80071">∫</m:mo>
															<m:mrow id="id80073">
																<m:mi id="id80074">X</m:mi>
																<m:mo id="id80076">⁣</m:mo>
																<m:mo id="id80079">/</m:mo>
																<m:mo id="id80081">⁣</m:mo>
																<m:mrow id="id80083">
																	<m:none id="id80084"/>
																	<m:mo id="id80085">/</m:mo>
																	<m:mi id="id80088">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mrow id="id80090">
															<m:mi id="id80091">c</m:mi>
															<m:mo id="id80093">⁢</m:mo>
															<m:mi id="id80095">h</m:mi>
														</m:mrow>
														<m:mo id="id80097">(</m:mo>
														<m:mover id="id80100" accent="true">
															<m:mi id="id80103">V</m:mi>
															<m:mo id="id80105">~</m:mo>
														</m:mover>
														<m:mo id="id80107">)</m:mo>
														<m:mo id="id80109">⌣</m:mo>
														<m:mrow id="id80112">
															<m:mi id="id80113">T</m:mi>
															<m:mo id="id80115">⁢</m:mo>
															<m:mi id="id80117">d</m:mi>
														</m:mrow>
														<m:mi id="id80119">T</m:mi>
														<m:mo id="id80122">(</m:mo>
														<m:mi id="id80124">X</m:mi>
														<m:mo id="id80126">/</m:mo>
														<m:mo id="id80128">/</m:mo>
														<m:mi id="id80130">T</m:mi>
														<m:mo id="id80132">)</m:mo>
														<m:mo id="id80134">⌣</m:mo>
														<m:mo id="id80137" mathsize="1.1em">(</m:mo>
														<m:mrow id="id80141">
															<m:mi id="id80142">c</m:mi>
															<m:mo id="id80144">⁢</m:mo>
															<m:mi id="id80147">h</m:mi>
														</m:mrow>
														<m:mo id="id80149">(</m:mo>
														<m:msup id="id80151">
															<m:mi id="id80152" mathvariant="normal">Λ</m:mi>
															<m:mtext id="id80157">even</m:mtext>
														</m:msup>
														<m:mi id="id80159">E</m:mi>
														<m:mo id="id80161">)</m:mo>
														<m:mo id="id80163">-</m:mo>
														<m:mrow id="id80166">
															<m:mi id="id80167">c</m:mi>
															<m:mo id="id80169">⁢</m:mo>
															<m:mi id="id80171">h</m:mi>
														</m:mrow>
														<m:mo id="id80173">(</m:mo>
														<m:msup id="id80175">
															<m:mi id="id80176" mathvariant="normal">Λ</m:mi>
															<m:mtext id="id80181">odd</m:mtext>
														</m:msup>
														<m:mi id="id80183">E</m:mi>
														<m:mo id="id80185">)</m:mo>
														<m:mo id="id80188" mathsize="1.1em">)</m:mo>
														<m:mo id="id80192">=</m:mo>
														<m:msub id="id80194">
															<m:mo id="id80195">∫</m:mo>
															<m:mrow id="id80198">
																<m:mi id="id80199">X</m:mi>
																<m:mo id="id80201">⁣</m:mo>
																<m:mo id="id80203">/</m:mo>
																<m:mo id="id80205">⁣</m:mo>
																<m:mrow id="id80208">
																	<m:none id="id80209"/>
																	<m:mo id="id80210">/</m:mo>
																	<m:mi id="id80212">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msub>
														<m:mo id="id80214" mathsize="1.1em">(</m:mo>
														<m:mrow id="id80218">
															<m:mi id="id80220">c</m:mi>
															<m:mo id="id80222">⁢</m:mo>
															<m:mi id="id80224">h</m:mi>
														</m:mrow>
														<m:mo id="id80226">(</m:mo>
														<m:mover id="id80228" accent="true">
															<m:mi id="id80232">V</m:mi>
															<m:mo id="id80234">~</m:mo>
														</m:mover>
														<m:mo id="id80236">⊗</m:mo>
														<m:msup id="id80238">
															<m:mi id="id80239" mathvariant="normal">Λ</m:mi>
															<m:mtext id="id80244">even</m:mtext>
														</m:msup>
														<m:mi id="id80246">E</m:mi>
														<m:mo id="id80249">)</m:mo>
														<m:mo id="id80251">-</m:mo>
														<m:mo id="id80253">(</m:mo>
														<m:mrow id="id80255">
															<m:mi id="id80256">c</m:mi>
															<m:mo id="id80258">⁢</m:mo>
															<m:mi id="id80261">h</m:mi>
														</m:mrow>
														<m:mo id="id80263">(</m:mo>
														<m:mover id="id80265" accent="true">
															<m:mi id="id80268">V</m:mi>
															<m:mo id="id80270">~</m:mo>
														</m:mover>
														<m:mo id="id80272">⊗</m:mo>
														<m:msup id="id80275">
															<m:mi id="id80276" mathvariant="normal">Λ</m:mi>
															<m:mtext id="id80281">odd</m:mtext>
														</m:msup>
														<m:mi id="id80283">E</m:mi>
														<m:mo id="id80285">)</m:mo>
														<m:mo id="id80287" mathsize="1.1em">)</m:mo>
														<m:mo id="id80291">⌣</m:mo>
														<m:mrow id="id80294">
															<m:mi id="id80295">T</m:mi>
															<m:mo id="id80297">⁢</m:mo>
															<m:mi id="id80299">d</m:mi>
														</m:mrow>
														<m:mi id="id80302">T</m:mi>
														<m:mo id="id80304">(</m:mo>
														<m:mi id="id80306">X</m:mi>
														<m:mo id="id80308">/</m:mo>
														<m:mo id="id80310">/</m:mo>
														<m:mi id="id80312">T</m:mi>
														<m:mo id="id80314">)</m:mo>
														<m:mo id="id80316">=</m:mo>
														<m:msup id="id80319">
															<m:mrow id="id80320">
																<m:mi id="id80321">i</m:mi>
																<m:mo id="id80323">⁢</m:mo>
																<m:mi id="id80325">n</m:mi>
																<m:mo id="id80327">⁢</m:mo>
																<m:mi id="id80330">d</m:mi>
																<m:mo id="id80332">⁢</m:mo>
																<m:mi id="id80334">e</m:mi>
																<m:mo id="id80336">⁢</m:mo>
																<m:mi id="id80339">x</m:mi>
															</m:mrow>
															<m:mrow id="id80341">
																<m:mi id="id80342">X</m:mi>
																<m:mo id="id80344">⁣</m:mo>
																<m:mo id="id80347">/</m:mo>
																<m:mo id="id80349">⁣</m:mo>
																<m:mrow id="id80351">
																	<m:none id="id80352"/>
																	<m:mo id="id80353">/</m:mo>
																	<m:mi id="id80355">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msup>
														<m:msub id="id80358">
															<m:mi id="id80359">D</m:mi>
															<m:mrow id="id80361">
																<m:mrow id="id80362">
																	<m:mover id="id80363" accent="true">
																		<m:mi id="id80366">V</m:mi>
																		<m:mo id="id80368">~</m:mo>
																	</m:mover>
																	<m:mo id="id80370">⊗</m:mo>
																	<m:msup id="id80373">
																		<m:mi id="id80374" mathvariant="normal">Λ</m:mi>
																		<m:mtext id="id80379">even</m:mtext>
																	</m:msup>
																</m:mrow>
																<m:mo id="id80381">⁢</m:mo>
																<m:mi id="id80383">E</m:mi>
															</m:mrow>
														</m:msub>
														<m:mo id="id80386">-</m:mo>
														<m:msup id="id80388">
															<m:mrow id="id80389">
																<m:mi id="id80390">i</m:mi>
																<m:mo id="id80392">⁢</m:mo>
																<m:mi id="id80394">n</m:mi>
																<m:mo id="id80396">⁢</m:mo>
																<m:mi id="id80399">d</m:mi>
																<m:mo id="id80401">⁢</m:mo>
																<m:mi id="id80403">e</m:mi>
																<m:mo id="id80406">⁢</m:mo>
																<m:mi id="id80408">x</m:mi>
															</m:mrow>
															<m:mrow id="id80410">
																<m:mi id="id80411">X</m:mi>
																<m:mo id="id80413">⁣</m:mo>
																<m:mo id="id80416">/</m:mo>
																<m:mo id="id80418">⁣</m:mo>
																<m:mrow id="id80420">
																	<m:none id="id80421"/>
																	<m:mo id="id80422">/</m:mo>
																	<m:mi id="id80424">T</m:mi>
																</m:mrow>
															</m:mrow>
														</m:msup>
														<m:msub id="id80427">
															<m:mi id="id80428">D</m:mi>
															<m:mrow id="id80430">
																<m:mrow id="id80431">
																	<m:mover id="id80432" accent="true">
																		<m:mi id="id80435">V</m:mi>
																		<m:mo id="id80437">~</m:mo>
																	</m:mover>
																	<m:mo id="id80440">⊗</m:mo>
																	<m:msup id="id80442">
																		<m:mi id="id80443" mathvariant="normal">Λ</m:mi>
																		<m:mtext id="id80448">odd</m:mtext>
																	</m:msup>
																</m:mrow>
																<m:mo id="id80450">⁢</m:mo>
																<m:mi id="id80452">E</m:mi>
															</m:mrow>
														</m:msub>
														<m:mo id="id80454">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id80456" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id80461" class="eqpad"/>
									</tr>

								</table>
								<p id="id80467" style="" class="p">Note finally that the formula we have derived is stated in terms of a
choice of positive roots, but the proof does not depend on any
properties of that choice, and hence the result holds for
any choice of positive roots.
∎</p>
							</div>

						</div>

					</div>

					<div class="section" id="S5">
						<h2 id="id80488" class="title section-title">
							<span id="id80491" class="tag">5 </span>Characteristic numbers</h2>
						<div class="para" id="S5.p1">
							<p id="id80505" style="" class="p">Using the <m:math id="id80511" display="inline">
									<m:semantics id="id80514">
										<m:mi id="id80515">K</m:mi>
										<m:annotation-xml id="id80518" encoding="MathML-Content">
											<m:ci id="id80521">K</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-theoretic arguments from section <a id="id80526" href="#S4" title="4 The index formula ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
									<span id="id80534" style="" class="text tag">4</span>
								</a>, it is a
simple matter to derive formulæ which express various characteristic
numbers of <m:math id="id80542" display="inline">
									<m:semantics id="id80545">
										<m:mrow id="id80546">
											<m:mi id="id80547">X</m:mi>
											<m:mo id="id80549">/</m:mo>
											<m:mo id="id80552">/</m:mo>
											<m:mi id="id80554">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id80556" encoding="MathML-Content">
											<m:ci id="id80559">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> in terms of characteristic numbers of <m:math id="id80563" display="inline">
									<m:semantics id="id80566">
										<m:mrow id="id80567">
											<m:mi id="id80568">X</m:mi>
											<m:mo id="id80570">/</m:mo>
											<m:mo id="id80572">/</m:mo>
											<m:mi id="id80575">T</m:mi>
										</m:mrow>
										<m:annotation-xml id="id80577" encoding="MathML-Content">
											<m:ci id="id80580">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
Recall that the tangent bundles <m:math id="id80584" display="inline">
									<m:semantics id="id80587">
										<m:mrow id="id80588">
											<m:mi id="id80589">T</m:mi>
											<m:mo id="id80591">(</m:mo>
											<m:mi id="id80593">X</m:mi>
											<m:mo id="id80596">/</m:mo>
											<m:mo id="id80598">/</m:mo>
											<m:mi id="id80600">G</m:mi>
											<m:mo id="id80602">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id80604" encoding="MathML-Content">
											<m:ci id="id80607">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id80611" display="inline">
									<m:semantics id="id80614">
										<m:mrow id="id80615">
											<m:mi id="id80616">T</m:mi>
											<m:mo id="id80618">(</m:mo>
											<m:mi id="id80621">X</m:mi>
											<m:mo id="id80623">/</m:mo>
											<m:mo id="id80625">/</m:mo>
											<m:mi id="id80627">T</m:mi>
											<m:mo id="id80629">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id80631" encoding="MathML-Content">
											<m:ci id="id80635">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> can be
considered as complex vector bundles in an essentially unique way, by
taking almost complex structures compatible with their symplectic forms.</p>
						</div>

						<div class="para" id="S5.p2">
							<p id="id80648" style="" class="p">In the proof of the index formula, we used the fact that
								<m:math id="id80655" display="inline">
									<m:semantics id="id80658">
										<m:mrow id="id80659">
											<m:mo id="id80660">[</m:mo>
											<m:mi id="id80662">T</m:mi>
											<m:mo id="id80664">(</m:mo>
											<m:mi id="id80667">X</m:mi>
											<m:mo id="id80669">/</m:mo>
											<m:mo id="id80671">/</m:mo>
											<m:mi id="id80673">T</m:mi>
											<m:mo id="id80675">)</m:mo>
											<m:mo id="id80677">]</m:mo>
											<m:mo id="id80679">-</m:mo>
											<m:mo id="id80682">[</m:mo>
											<m:mi id="id80684">E</m:mi>
											<m:mo id="id80686">⊕</m:mo>
											<m:msup id="id80688">
												<m:mi id="id80689">E</m:mi>
												<m:mo id="id80691">*</m:mo>
											</m:msup>
											<m:mo id="id80694">]</m:mo>
											<m:mo id="id80696">∈</m:mo>
											<m:mi id="id80698">K</m:mi>
											<m:mo id="id80700">(</m:mo>
											<m:mi id="id80702">X</m:mi>
											<m:mo id="id80704">/</m:mo>
											<m:mo id="id80707">/</m:mo>
											<m:mi id="id80709">T</m:mi>
											<m:mo id="id80711">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id80713" encoding="MathML-Content"/>
									</m:semantics>
								</m:math> is a lift of <m:math id="id80718" display="inline">
									<m:semantics id="id80721">
										<m:mrow id="id80722">
											<m:mo id="id80723">[</m:mo>
											<m:mi id="id80725">T</m:mi>
											<m:mo id="id80727">(</m:mo>
											<m:mi id="id80730">X</m:mi>
											<m:mo id="id80732">/</m:mo>
											<m:mo id="id80734">/</m:mo>
											<m:mi id="id80736">G</m:mi>
											<m:mo id="id80738">)</m:mo>
											<m:mo id="id80740">]</m:mo>
											<m:mo id="id80742">∈</m:mo>
											<m:mi id="id80745">K</m:mi>
											<m:mo id="id80747">(</m:mo>
											<m:mi id="id80749">X</m:mi>
											<m:mo id="id80751">/</m:mo>
											<m:mo id="id80753">/</m:mo>
											<m:mi id="id80755">G</m:mi>
											<m:mo id="id80758">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id80760" encoding="MathML-Content">
											<m:ci id="id80763">[</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. It follows that</p>
							<table class="equation" id="S5.Ex1">

								<tr valign="baseline" class="equation" id="S5.Ex1">
									<td id="id80783" class="eqpad"/>
									<td id="id80788" nowrap="yes" align="center" colspan="1">
										<m:math id="id80795" display="block">
											<m:semantics id="id80798">
												<m:mfrac id="id80799">
													<m:mrow id="id80800">
														<m:mi id="id80801">c</m:mi>
														<m:mo id="id80804">(</m:mo>
														<m:mi id="id80806">T</m:mi>
														<m:mo id="id80808">(</m:mo>
														<m:mi id="id80810">X</m:mi>
														<m:mo id="id80812">/</m:mo>
														<m:mo id="id80814">/</m:mo>
														<m:mi id="id80816">T</m:mi>
														<m:mo id="id80818">)</m:mo>
														<m:mo id="id80821">)</m:mo>
													</m:mrow>
													<m:mrow id="id80823">
														<m:mrow id="id80824">
															<m:mi id="id80825">c</m:mi>
															<m:mo id="id80827">⁢</m:mo>
															<m:mfenced id="id80829" open="(" close=")">
																<m:mi id="id80834">E</m:mi>
															</m:mfenced>
														</m:mrow>
														<m:mo id="id80837">⌣</m:mo>
														<m:mrow id="id80839">
															<m:mi id="id80840">c</m:mi>
															<m:mo id="id80842">⁢</m:mo>
															<m:mfenced id="id80845" open="(" close=")">
																<m:msup id="id80850">
																	<m:mi id="id80851">E</m:mi>
																	<m:mo id="id80853">*</m:mo>
																</m:msup>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
												</m:mfrac>
												<m:annotation-xml id="id80855" encoding="MathML-Content">
													<m:apply id="id80858">
														<m:divide id="id80859"/>
														<m:mtext id="id80860">c(T(X//T))</m:mtext>
														<m:apply id="id80863">
															<m:ci id="id80864">⌣</m:ci>
															<m:apply id="id80866">
																<m:times id="id80867"/>
																<m:ci id="id80868">c</m:ci>
																<m:ci id="id80871">E</m:ci>
															</m:apply>
															<m:apply id="id80873">
																<m:times id="id80874"/>
																<m:ci id="id80875">c</m:ci>
																<m:apply id="id80877">
																	<m:csymbol id="id80878" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id80883">E</m:ci>
																	<m:times id="id80885"/>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</td>
									<td id="id80887" class="eqpad"/>
								</tr>

							</table>
							<p id="id80894" style="" class="p">is a lift of the total Chern class <m:math id="id80900" display="inline">
									<m:semantics id="id80903">
										<m:mrow id="id80904">
											<m:mi id="id80906">c</m:mi>
											<m:mo id="id80908">(</m:mo>
											<m:mi id="id80910">T</m:mi>
											<m:mo id="id80912">(</m:mo>
											<m:mi id="id80914">X</m:mi>
											<m:mo id="id80916">/</m:mo>
											<m:mo id="id80918">/</m:mo>
											<m:mi id="id80920">G</m:mi>
											<m:mo id="id80923">)</m:mo>
											<m:mo id="id80925">)</m:mo>
										</m:mrow>
										<m:annotation-xml id="id80927" encoding="MathML-Content">
											<m:ci id="id80930">c</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Hence, applying the
integration formula, we get the following formula for the Euler
characteristic of <m:math id="id80934" display="inline">
									<m:semantics id="id80938">
										<m:mrow id="id80939">
											<m:mi id="id80940">X</m:mi>
											<m:mo id="id80942">/</m:mo>
											<m:mo id="id80944">/</m:mo>
											<m:mi id="id80946">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id80948" encoding="MathML-Content">
											<m:ci id="id80952">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
							</p>
							<table class="equation" id="S5.E1">

								<tr valign="baseline" class="equation" id="S5.E1">
									<td id="id80971" class="eqpad"/>
									<td id="id80975" nowrap="yes" align="center" colspan="1">
										<m:math id="id80982" display="block">
											<m:semantics id="id80986">
												<m:mrow id="id80987">
													<m:mi id="id80988">χ</m:mi>
													<m:mo id="id80990">(</m:mo>
													<m:mi id="id80992">X</m:mi>
													<m:mo id="id80994">/</m:mo>
													<m:mo id="id80997">/</m:mo>
													<m:mi id="id80999">G</m:mi>
													<m:mo id="id81001">)</m:mo>
													<m:mo id="id81003">=</m:mo>
													<m:mfrac id="id81005">
														<m:mn id="id81006">1</m:mn>
														<m:mfenced id="id81008" open="|" close="|">
															<m:mi id="id81013">W</m:mi>
														</m:mfenced>
													</m:mfrac>
													<m:msub id="id81016">
														<m:mo id="id81017">∫</m:mo>
														<m:mrow id="id81019">
															<m:mi id="id81020">X</m:mi>
															<m:mo id="id81022">⁣</m:mo>
															<m:mo id="id81025">/</m:mo>
															<m:mo id="id81027">⁣</m:mo>
															<m:mrow id="id81029">
																<m:none id="id81030"/>
																<m:mo id="id81031">/</m:mo>
																<m:mi id="id81033">T</m:mi>
															</m:mrow>
														</m:mrow>
													</m:msub>
													<m:mi id="id81036">c</m:mi>
													<m:mo id="id81038">(</m:mo>
													<m:mi id="id81040">T</m:mi>
													<m:mo id="id81042">(</m:mo>
													<m:mi id="id81044">X</m:mi>
													<m:mo id="id81046">/</m:mo>
													<m:mo id="id81048">/</m:mo>
													<m:mi id="id81050">T</m:mi>
													<m:mo id="id81053">)</m:mo>
													<m:mo id="id81055">)</m:mo>
													<m:mo id="id81057">⌣</m:mo>
													<m:munder id="id81059">
														<m:mo id="id81060" movablelimits="false">∏</m:mo>
														<m:mrow id="id81065">
															<m:mi id="id81066">α</m:mi>
															<m:mo id="id81068">∈</m:mo>
															<m:mi id="id81071" mathvariant="normal">Δ</m:mi>
														</m:mrow>
													</m:munder>
													<m:mfrac id="id81076">
														<m:mrow id="id81077">
															<m:mi id="id81078">e</m:mi>
															<m:mo id="id81080">⁢</m:mo>
															<m:mfenced id="id81082" open="(" close=")">
																<m:mi id="id81087">α</m:mi>
															</m:mfenced>
														</m:mrow>
														<m:mrow id="id81090">
															<m:mn id="id81091">1</m:mn>
															<m:mo id="id81093">+</m:mo>
															<m:mrow id="id81095">
																<m:mi id="id81096">e</m:mi>
																<m:mo id="id81098">⁢</m:mo>
																<m:mfenced id="id81101" open="(" close=")">
																	<m:mi id="id81106">α</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mfrac>
													<m:mo id="id81108">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id81110" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</td>
									<td id="id81115" class="eqpad"/>
									<td id="id81118" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id81129" class="refnum">5.1</span>)</td>
								</tr>

							</table>
							<p id="id81137" style="" class="p">(We are using the fact that the top Chern class equals the Euler
class.)</p>
						</div>

						<div class="para" id="S5.p3">
							<p id="id81153" style="" class="p">Similarly, taking the <m:math id="id81159" display="inline">
									<m:semantics id="id81163">
										<m:mi id="id81164">L</m:mi>
										<m:annotation-xml id="id81166" encoding="MathML-Content">
											<m:ci id="id81169">L</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-class,
							</p>
							<table class="equation" id="S5.E2">

								<tr valign="baseline" class="equation" id="S5.E2">
									<td id="id81189" class="eqpad"/>
									<td id="id81194" nowrap="yes" align="center" colspan="1">
										<m:math id="id81201" display="block">
											<m:semantics id="id81204">
												<m:mrow id="id81206">
													<m:mo id="id81207">signature</m:mo>
													<m:mo id="id81209">(</m:mo>
													<m:mi id="id81211">X</m:mi>
													<m:mo id="id81213">/</m:mo>
													<m:mo id="id81215">/</m:mo>
													<m:mi id="id81218">G</m:mi>
													<m:mo id="id81220">)</m:mo>
													<m:mo id="id81222">=</m:mo>
													<m:mfrac id="id81224">
														<m:mn id="id81225">1</m:mn>
														<m:mfenced id="id81227" open="|" close="|">
															<m:mi id="id81232">W</m:mi>
														</m:mfenced>
													</m:mfrac>
													<m:msub id="id81234">
														<m:mo id="id81235">∫</m:mo>
														<m:mrow id="id81238">
															<m:mi id="id81239">X</m:mi>
															<m:mo id="id81241">⁣</m:mo>
															<m:mo id="id81243">/</m:mo>
															<m:mo id="id81246">⁣</m:mo>
															<m:mrow id="id81248">
																<m:none id="id81249"/>
																<m:mo id="id81250">/</m:mo>
																<m:mi id="id81252">T</m:mi>
															</m:mrow>
														</m:mrow>
													</m:msub>
													<m:mi id="id81254">L</m:mi>
													<m:mo id="id81256">(</m:mo>
													<m:mi id="id81259">T</m:mi>
													<m:mo id="id81261">(</m:mo>
													<m:mi id="id81263">X</m:mi>
													<m:mo id="id81265">/</m:mo>
													<m:mo id="id81267">/</m:mo>
													<m:mi id="id81269">T</m:mi>
													<m:mo id="id81271">)</m:mo>
													<m:mo id="id81274">)</m:mo>
													<m:mo id="id81276">⌣</m:mo>
													<m:munder id="id81278">
														<m:mo id="id81279" movablelimits="false">∏</m:mo>
														<m:mrow id="id81284">
															<m:mi id="id81285">α</m:mi>
															<m:mo id="id81287">∈</m:mo>
															<m:mi id="id81290" mathvariant="normal">Δ</m:mi>
														</m:mrow>
													</m:munder>
													<m:mo id="id81294">tanh</m:mo>
													<m:mi id="id81297">e</m:mi>
													<m:mo id="id81299">(</m:mo>
													<m:mi id="id81301">α</m:mi>
													<m:mo id="id81303">)</m:mo>
													<m:mo id="id81306">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id81308" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</td>
									<td id="id81312" class="eqpad"/>
									<td id="id81315" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id81326" class="refnum">5.2</span>)</td>
								</tr>

							</table>
						</div>

						<div class="para" id="S5.p4">
							<p id="id81341" style="" class="p">In general, for any ‘multiplicative characteristic class’ <m:math id="id81349" display="inline">
									<m:semantics id="id81352">
										<m:mi id="id81353">m</m:mi>
										<m:annotation-xml id="id81355" encoding="MathML-Content">
											<m:ci id="id81359">m</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> (see
Hirzebruch <cite id="id81362" class="cite">[<a id="id81367" href="#bib.bib8" title="" class="ref">8</a>, section 1]</cite>, or Milnor and
Stasheff <cite id="id81378" class="cite">[<a id="id81383" href="#bib.bib18" title="" class="ref">18</a>, section 19]</cite>), we have</p>
							<table class="equation" id="S5.E3">

								<tr valign="baseline" class="equation" id="S5.E3">
									<td id="id81410" class="eqpad"/>
									<td id="id81415" nowrap="yes" align="center" colspan="1">
										<m:math id="id81422" display="block">
											<m:semantics id="id81426">
												<m:mrow id="id81427">
													<m:msub id="id81428">
														<m:mo id="id81429">∫</m:mo>
														<m:mrow id="id81431">
															<m:mi id="id81432">X</m:mi>
															<m:mo id="id81434">⁣</m:mo>
															<m:mo id="id81437">/</m:mo>
															<m:mo id="id81439">⁣</m:mo>
															<m:mrow id="id81441">
																<m:none id="id81442"/>
																<m:mo id="id81443">/</m:mo>
																<m:mi id="id81446">G</m:mi>
															</m:mrow>
														</m:mrow>
													</m:msub>
													<m:mi id="id81448">m</m:mi>
													<m:mo id="id81450">(</m:mo>
													<m:mi id="id81452">T</m:mi>
													<m:mo id="id81454">(</m:mo>
													<m:mi id="id81456">X</m:mi>
													<m:mo id="id81458">/</m:mo>
													<m:mo id="id81460">/</m:mo>
													<m:mi id="id81463">G</m:mi>
													<m:mo id="id81465">)</m:mo>
													<m:mo id="id81467">)</m:mo>
													<m:mo id="id81469">=</m:mo>
													<m:mfrac id="id81471">
														<m:mn id="id81472">1</m:mn>
														<m:mfenced id="id81474" open="|" close="|">
															<m:mi id="id81479">W</m:mi>
														</m:mfenced>
													</m:mfrac>
													<m:msub id="id81482">
														<m:mo id="id81483">∫</m:mo>
														<m:mrow id="id81485">
															<m:mi id="id81486">X</m:mi>
															<m:mo id="id81488">⁣</m:mo>
															<m:mo id="id81491">/</m:mo>
															<m:mo id="id81493">⁣</m:mo>
															<m:mrow id="id81495">
																<m:none id="id81496"/>
																<m:mo id="id81497">/</m:mo>
																<m:mi id="id81499">T</m:mi>
															</m:mrow>
														</m:mrow>
													</m:msub>
													<m:mfrac id="id81502">
														<m:mrow id="id81503">
															<m:mi id="id81504">m</m:mi>
															<m:mo id="id81506">(</m:mo>
															<m:mi id="id81508">T</m:mi>
															<m:mo id="id81510">(</m:mo>
															<m:mi id="id81512">X</m:mi>
															<m:mo id="id81514">/</m:mo>
															<m:mo id="id81516">/</m:mo>
															<m:mi id="id81519">T</m:mi>
															<m:mo id="id81521">)</m:mo>
															<m:mo id="id81523">)</m:mo>
														</m:mrow>
														<m:mrow id="id81525">
															<m:mrow id="id81526">
																<m:mi id="id81527">m</m:mi>
																<m:mo id="id81529">⁢</m:mo>
																<m:mfenced id="id81532" open="(" close=")">
																	<m:mi id="id81537">E</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id81539">⌣</m:mo>
															<m:mrow id="id81541">
																<m:mi id="id81542">m</m:mi>
																<m:mo id="id81544">⁢</m:mo>
																<m:mfenced id="id81547" open="(" close=")">
																	<m:msup id="id81552">
																		<m:mi id="id81553">E</m:mi>
																		<m:mo id="id81555">*</m:mo>
																	</m:msup>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mfrac>
													<m:mo id="id81557">⌣</m:mo>
													<m:munder id="id81560">
														<m:mo id="id81561" movablelimits="false">∏</m:mo>
														<m:mrow id="id81565">
															<m:mi id="id81566">α</m:mi>
															<m:mo id="id81569">∈</m:mo>
															<m:mi id="id81571" mathvariant="normal">Δ</m:mi>
														</m:mrow>
													</m:munder>
													<m:mi id="id81576">e</m:mi>
													<m:mo id="id81578">(</m:mo>
													<m:mi id="id81580">α</m:mi>
													<m:mo id="id81583">)</m:mo>
													<m:mo id="id81585">.</m:mo>
												</m:mrow>
												<m:annotation-xml id="id81587" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</td>
									<td id="id81591" class="eqpad"/>
									<td id="id81595" rowspan="1" nowrap="yes" valign="middle" align="right">(<span id="id81605" class="refnum">5.3</span>)</td>
								</tr>

							</table>
						</div>

					</div>

					<div class="section" id="S6">
						<h2 id="id81622" class="title section-title">
							<span id="id81625" class="tag">6 </span>Generalizations</h2>
						<div class="para" id="S6.p1">
							<p id="id81639" style="" class="p">In the previous sections of this paper we have assumed that both
								<m:math id="id81646" display="inline">
									<m:semantics id="id81649">
										<m:mrow id="id81650">
											<m:msubsup id="id81651">
												<m:mi id="id81652">μ</m:mi>
												<m:mi id="id81654">G</m:mi>
												<m:mrow id="id81657">
													<m:mo id="id81658">-</m:mo>
													<m:mn id="id81660">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id81662">⁢</m:mo>
											<m:mfenced id="id81664" open="(" close=")">
												<m:mn id="id81669">0</m:mn>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id81672" encoding="MathML-Content">
											<m:apply id="id81675">
												<m:times id="id81676"/>
												<m:apply id="id81677">
													<m:csymbol id="id81678" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id81683">
														<m:csymbol id="id81684" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id81688">μ</m:ci>
														<m:ci id="id81691">G</m:ci>
													</m:apply>
													<m:apply id="id81693">
														<m:minus id="id81694"/>
														<m:cn id="id81695">1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn id="id81697">0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id81701" display="inline">
									<m:semantics id="id81704">
										<m:mrow id="id81705">
											<m:msubsup id="id81706">
												<m:mi id="id81707">μ</m:mi>
												<m:mi id="id81710">T</m:mi>
												<m:mrow id="id81712">
													<m:mo id="id81713">-</m:mo>
													<m:mn id="id81715">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id81717">⁢</m:mo>
											<m:mfenced id="id81720" open="(" close=")">
												<m:mn id="id81725">0</m:mn>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id81727" encoding="MathML-Content">
											<m:apply id="id81730">
												<m:times id="id81731"/>
												<m:apply id="id81732">
													<m:csymbol id="id81733" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id81738">
														<m:csymbol id="id81739" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id81744">μ</m:ci>
														<m:ci id="id81746">T</m:ci>
													</m:apply>
													<m:apply id="id81748">
														<m:minus id="id81749"/>
														<m:cn id="id81750">1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn id="id81752">0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> are compact manifolds on which the
respective <m:math id="id81756" display="inline">
									<m:semantics id="id81760">
										<m:mi id="id81761">G</m:mi>
										<m:annotation-xml id="id81763" encoding="MathML-Content">
											<m:ci id="id81766">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>- and <m:math id="id81770" display="inline">
									<m:semantics id="id81773">
										<m:mi id="id81774">T</m:mi>
										<m:annotation-xml id="id81776" encoding="MathML-Content">
											<m:ci id="id81779">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-actions are free.
In this section we show how to remove some of these assumptions.
We will keep the assumption that <m:math id="id81784" display="inline">
									<m:semantics id="id81787">
										<m:msub id="id81788">
											<m:mi id="id81789">μ</m:mi>
											<m:mi id="id81792">G</m:mi>
										</m:msub>
										<m:annotation-xml id="id81794" encoding="MathML-Content">
											<m:apply id="id81797">
												<m:csymbol id="id81798" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id81803">μ</m:ci>
												<m:ci id="id81805">G</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a proper map,
having <m:math id="id81809" display="inline">
									<m:semantics id="id81812">
										<m:mn id="id81813">0</m:mn>
										<m:annotation-xml id="id81815" encoding="MathML-Content">
											<m:cn id="id81819">0</m:cn>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as a regular value. From this it follows that <m:math id="id81822" display="inline">
									<m:semantics id="id81826">
										<m:mrow id="id81827">
											<m:msubsup id="id81828">
												<m:mi id="id81829">μ</m:mi>
												<m:mi id="id81831">G</m:mi>
												<m:mrow id="id81833">
													<m:mo id="id81834">-</m:mo>
													<m:mn id="id81837">1</m:mn>
												</m:mrow>
											</m:msubsup>
											<m:mo id="id81839">⁢</m:mo>
											<m:mfenced id="id81841" open="(" close=")">
												<m:mn id="id81846">0</m:mn>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id81848" encoding="MathML-Content">
											<m:apply id="id81852">
												<m:times id="id81853"/>
												<m:apply id="id81854">
													<m:csymbol id="id81855" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id81860">
														<m:csymbol id="id81861" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id81865">μ</m:ci>
														<m:ci id="id81868">G</m:ci>
													</m:apply>
													<m:apply id="id81870">
														<m:minus id="id81871"/>
														<m:cn id="id81872">1</m:cn>
													</m:apply>
												</m:apply>
												<m:cn id="id81874">0</m:cn>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
is a compact manifold, on which the <m:math id="id81878" display="inline">
									<m:semantics id="id81881">
										<m:mi id="id81882">G</m:mi>
										<m:annotation-xml id="id81884" encoding="MathML-Content">
											<m:ci id="id81888">G</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-action is locally free, and
hence that <m:math id="id81891" display="inline">
									<m:semantics id="id81895">
										<m:mrow id="id81896">
											<m:mi id="id81897">X</m:mi>
											<m:mo id="id81899">/</m:mo>
											<m:mo id="id81901">/</m:mo>
											<m:mi id="id81903">G</m:mi>
										</m:mrow>
										<m:annotation-xml id="id81905" encoding="MathML-Content">
											<m:ci id="id81909">X</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is a compact symplectic orbifold.</p>
						</div>

						<div class="subsection" id="S6.SS1">
							<h3 id="id81921" class="title subsection-title">
								<span id="id81924" class="tag">6.1 </span>The case in which <m:math id="id81930" display="inline">
									<m:semantics id="id81933">
										<m:msub id="id81934">
											<m:mi id="id81936">μ</m:mi>
											<m:mi id="id81938">T</m:mi>
										</m:msub>
										<m:annotation-xml id="id81940" encoding="MathML-Content">
											<m:apply id="id81943">
												<m:csymbol id="id81944" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id81949">μ</m:ci>
												<m:ci id="id81952">T</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is proper and has <m:math id="id81955" display="inline">
									<m:semantics id="id81958">
										<m:mn id="id81959">0</m:mn>
										<m:annotation-xml id="id81962" encoding="MathML-Content">
											<m:cn id="id81965">0</m:cn>
										</m:annotation-xml>
									</m:semantics>
								</m:math> as a regular
value</h3>
							<div class="para" id="S6.SS1.p1">
								<p id="id81976" style="" class="p">If <m:math id="id81982" display="inline">
										<m:semantics id="id81986">
											<m:msub id="id81987">
												<m:mi id="id81988">μ</m:mi>
												<m:mi id="id81990">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id81992" encoding="MathML-Content">
												<m:apply id="id81996">
													<m:csymbol id="id81997" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id82001">μ</m:ci>
													<m:ci id="id82004">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is proper and has <m:math id="id82007" display="inline">
										<m:semantics id="id82010">
											<m:mn id="id82012">0</m:mn>
											<m:annotation-xml id="id82014" encoding="MathML-Content">
												<m:cn id="id82017">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> as a regular value, then <m:math id="id82021" display="inline">
										<m:semantics id="id82024">
											<m:mrow id="id82025">
												<m:mi id="id82026">X</m:mi>
												<m:mo id="id82028">/</m:mo>
												<m:mo id="id82030">/</m:mo>
												<m:mi id="id82032">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id82035" encoding="MathML-Content">
												<m:ci id="id82038">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a
compact symplectic orbifold.
The arguments in
sections <a id="id82044" href="#S1" title="1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id82042" style="" class="text tag">1</span>
									</a>–<a id="id82060" href="#S3" title="3 The relationship between the cohomology rings ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id82058" style="" class="text tag">3</span>
									</a>, in which we proved the
integration formula and the formula relating the cohomology rings, go
through with straightforward modifications, which we now describe.</p>
							</div>

							<div class="para" id="S6.SS1.p2">
								<p id="id82084" style="" class="p">In the main topological result, proposition <a id="id82093" href="#S1.E2" title="Proposition 1.2 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id82091" style="" class="text tag">1.2</span>
									</a>, the
line bundles <m:math id="id82108" display="inline">
										<m:semantics id="id82111">
											<m:msub id="id82112">
												<m:mi id="id82113">L</m:mi>
												<m:mi id="id82115">α</m:mi>
											</m:msub>
											<m:annotation-xml id="id82117" encoding="MathML-Content">
												<m:apply id="id82121">
													<m:csymbol id="id82122" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id82126">L</m:ci>
													<m:ci id="id82128">α</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, as well as the normal bundle and the fibering
must all be replaced by their orbifold equivalents.  (In the companion
paper to this one <cite id="id82134" class="cite">[<a id="id82139" href="#bib.bib15" title="" class="ref">15</a>]</cite>, appendix A summarizes the main
topological and cohomological properties of orbifolds, orbifold vector
bundles, and orbifold fibre bundles, including describing how
integration over the fibre goes over to that case.)</p>
							</div>

							<div class="para" id="S6.SS1.p3">
								<p id="id82162" style="" class="p">The classes <m:math id="id82168" display="inline">
										<m:semantics id="id82172">
											<m:mrow id="id82173">
												<m:mi id="id82174">e</m:mi>
												<m:mo id="id82176">⁢</m:mo>
												<m:mfenced id="id82178" open="(" close=")">
													<m:mi id="id82183">α</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id82186" encoding="MathML-Content">
												<m:apply id="id82189">
													<m:times id="id82190"/>
													<m:ci id="id82191">e</m:ci>
													<m:ci id="id82193">α</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> are well-defined rational cohomology classes, and
theorem A extends to this case unchanged (rational Poincaré duality
holds for compact oriented orbifolds). Theorem B must be modified
to take into account the existence of global finite stabilizers, and
becomes</p>
							</div>

							<div class="theorem theoremBp" id="ThmtheoremBpx1">
								<h6 id="id82210" class="title theorem-title bold">
									<span id="id82214" class="tag">Theorem B<m:math id="id82218" display="inline">
											<m:semantics id="id82221">
												<m:msup id="id82222">
													<m:mi id="id82223"/>
													<m:mo id="id82224" mathvariant="normal">′</m:mo>
												</m:msup>
												<m:annotation-xml id="id82229" encoding="MathML-Content">
													<m:apply id="id82232">
														<m:ci id="id82234">′</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</span> (Integration formula).</h6>
								<div class="para" id="ThmtheoremBpx1.p1">
									<p id="id82245" style="" class="p">
										<span id="id82250" style="" class="text italic">If <m:math id="id82256" display="inline">
												<m:semantics id="id82260">
													<m:msub id="id82261">
														<m:mi id="id82262">μ</m:mi>
														<m:mi id="id82264">G</m:mi>
													</m:msub>
													<m:annotation-xml id="id82266" encoding="MathML-Content">
														<m:apply id="id82270">
															<m:csymbol id="id82271" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id82276">μ</m:ci>
															<m:ci id="id82278">G</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id82281" display="inline">
												<m:semantics id="id82285">
													<m:msub id="id82286">
														<m:mi id="id82287">μ</m:mi>
														<m:mi id="id82289">T</m:mi>
													</m:msub>
													<m:annotation-xml id="id82291" encoding="MathML-Content">
														<m:apply id="id82295">
															<m:csymbol id="id82296" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id82300">μ</m:ci>
															<m:ci id="id82303">T</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> are proper maps, both having <m:math id="id82306" display="inline">
												<m:semantics id="id82310">
													<m:mn id="id82311" mathvariant="normal">0</m:mn>
													<m:annotation-xml id="id82315" encoding="MathML-Content">
														<m:cn id="id82319">0</m:cn>
													</m:annotation-xml>
												</m:semantics>
											</m:math> as a regular
value, then for any class <m:math id="id82322" display="inline">
												<m:semantics id="id82326">
													<m:mrow id="id82327">
														<m:mi id="id82328">a</m:mi>
														<m:mo id="id82330" mathvariant="normal">∈</m:mo>
														<m:msup id="id82335">
															<m:mi id="id82336">H</m:mi>
															<m:mo id="id82338" mathvariant="normal">*</m:mo>
														</m:msup>
														<m:mo id="id82342" mathvariant="normal">(</m:mo>
														<m:mi id="id82347">X</m:mi>
														<m:mo id="id82349" mathvariant="normal">/</m:mo>
														<m:mo id="id82353" mathvariant="normal">/</m:mo>
														<m:mi id="id82358">G</m:mi>
														<m:mo id="id82360" mathvariant="normal">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id82364" encoding="MathML-Content"/>
												</m:semantics>
											</m:math> with lift <m:math id="id82369" display="inline">
												<m:semantics id="id82372">
													<m:mover id="id82373" accent="true">
														<m:mi id="id82376">a</m:mi>
														<m:mo id="id82379" mathvariant="normal">~</m:mo>
													</m:mover>
													<m:annotation-xml id="id82383" encoding="MathML-Content">
														<m:apply id="id82386">
															<m:ci id="id82387">~</m:ci>
															<m:ci id="id82390">a</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>,</span>
									</p>
									<table class="equation" id="S6.Ex1">

										<tr valign="baseline" class="equation" id="S6.Ex1">
											<td id="id82409" class="eqpad"/>
											<td id="id82414" nowrap="yes" align="center" colspan="1">
												<m:math id="id82421" display="block">
													<m:semantics id="id82424">
														<m:mrow id="id82426">
															<m:mrow id="id82427">
																<m:mrow id="id82428">
																	<m:msub id="id82429">
																		<m:mo id="id82430">∫</m:mo>
																		<m:mrow id="id82432">
																			<m:mi id="id82433">X</m:mi>
																			<m:mo id="id82435">⁣</m:mo>
																			<m:mo id="id82438">/</m:mo>
																			<m:mo id="id82440">⁣</m:mo>
																			<m:mrow id="id82442">
																				<m:none id="id82443"/>
																				<m:mo id="id82444">/</m:mo>
																				<m:mi id="id82447">G</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msub>
																	<m:mi id="id82449">a</m:mi>
																</m:mrow>
																<m:mo id="id82451">=</m:mo>
																<m:mrow id="id82453">
																	<m:mrow id="id82454">
																		<m:mfrac id="id82455">
																			<m:mn id="id82456">1</m:mn>
																			<m:mfenced id="id82458" open="|" close="|">
																				<m:mi id="id82463">W</m:mi>
																			</m:mfenced>
																		</m:mfrac>
																		<m:mo id="id82466">⋅</m:mo>
																		<m:mfrac id="id82468">
																			<m:mrow id="id82469">
																				<m:msub id="id82470">
																					<m:mi id="id82471">o</m:mi>
																					<m:mi id="id82473">T</m:mi>
																				</m:msub>
																				<m:mo id="id82475">⁢</m:mo>
																				<m:mfenced id="id82478" open="(" close=")">
																					<m:mrow id="id82483">
																						<m:msubsup id="id82484">
																							<m:mi id="id82485">μ</m:mi>
																							<m:mi id="id82487">T</m:mi>
																							<m:mrow id="id82490">
																								<m:mo id="id82491">-</m:mo>
																								<m:mn id="id82493">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id82495">⁢</m:mo>
																						<m:mfenced id="id82497" open="(" close=")">
																							<m:mn id="id82502">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																			<m:mrow id="id82504">
																				<m:msub id="id82506">
																					<m:mi id="id82507">o</m:mi>
																					<m:mi id="id82509">G</m:mi>
																				</m:msub>
																				<m:mo id="id82511">⁢</m:mo>
																				<m:mfenced id="id82513" open="(" close=")">
																					<m:mrow id="id82518">
																						<m:msubsup id="id82519">
																							<m:mi id="id82520">μ</m:mi>
																							<m:mi id="id82523">G</m:mi>
																							<m:mrow id="id82525">
																								<m:mo id="id82526">-</m:mo>
																								<m:mn id="id82528">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id82530">⁢</m:mo>
																						<m:mfenced id="id82533" open="(" close=")">
																							<m:mn id="id82538">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																		</m:mfrac>
																	</m:mrow>
																	<m:mo id="id82540">⁢</m:mo>
																	<m:mrow id="id82542">
																		<m:msub id="id82543">
																			<m:mo id="id82544">∫</m:mo>
																			<m:mrow id="id82547">
																				<m:mi id="id82548">X</m:mi>
																				<m:mo id="id82550">⁣</m:mo>
																				<m:mo id="id82552">/</m:mo>
																				<m:mo id="id82555">⁣</m:mo>
																				<m:mrow id="id82557">
																					<m:none id="id82558"/>
																					<m:mo id="id82559">/</m:mo>
																					<m:mi id="id82561">T</m:mi>
																				</m:mrow>
																			</m:mrow>
																		</m:msub>
																		<m:mover id="id82563" accent="true">
																			<m:mi id="id82567">a</m:mi>
																			<m:mo id="id82569">~</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mrow>
																<m:mo id="id82571">⌣</m:mo>
																<m:mi id="id82573">e</m:mi>
															</m:mrow>
															<m:mo id="id82576">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id82578" encoding="MathML-Content">
															<m:apply id="id82581">
																<m:ci id="id82582"/>
																<m:apply id="id82583">
																	<m:apply id="id82584">
																		<m:csymbol id="id82585" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id82590"/>
																		<m:apply id="id82591">
																			<m:list id="id82592"/>
																			<m:ci id="id82593">X</m:ci>
																			<m:divide id="id82595"/>
																			<m:apply id="id82596">
																				<m:divide id="id82597"/>
																				<m:ci id="id82598"/>
																				<m:ci id="id82600">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci id="id82602">a</m:ci>
																</m:apply>
																<m:eq id="id82604"/>
																<m:apply id="id82605">
																	<m:times id="id82606"/>
																	<m:apply id="id82607">
																		<m:ci id="id82608">⋅</m:ci>
																		<m:apply id="id82610">
																			<m:divide id="id82612"/>
																			<m:cn id="id82613">1</m:cn>
																			<m:apply id="id82615">
																				<m:abs id="id82616"/>
																				<m:ci id="id82617">W</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id82619">
																			<m:divide id="id82620"/>
																			<m:apply id="id82621">
																				<m:times id="id82622"/>
																				<m:apply id="id82623">
																					<m:csymbol id="id82624" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id82629">o</m:ci>
																					<m:ci id="id82631">T</m:ci>
																				</m:apply>
																				<m:apply id="id82633">
																					<m:times id="id82634"/>
																					<m:apply id="id82635">
																						<m:csymbol id="id82636" cd="ambiguous">superscript</m:csymbol>
																						<m:apply id="id82641">
																							<m:csymbol id="id82642" cd="ambiguous">subscript</m:csymbol>
																							<m:ci id="id82647">μ</m:ci>
																							<m:ci id="id82649">T</m:ci>
																						</m:apply>
																						<m:apply id="id82651">
																							<m:minus id="id82652"/>
																							<m:cn id="id82654">1</m:cn>
																						</m:apply>
																					</m:apply>
																					<m:cn id="id82656">0</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id82658">
																				<m:times id="id82659"/>
																				<m:apply id="id82660">
																					<m:csymbol id="id82661" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id82666">o</m:ci>
																					<m:ci id="id82668">G</m:ci>
																				</m:apply>
																				<m:apply id="id82670">
																					<m:times id="id82671"/>
																					<m:apply id="id82672">
																						<m:csymbol id="id82673" cd="ambiguous">superscript</m:csymbol>
																						<m:apply id="id82678">
																							<m:csymbol id="id82679" cd="ambiguous">subscript</m:csymbol>
																							<m:ci id="id82684">μ</m:ci>
																							<m:ci id="id82686">G</m:ci>
																						</m:apply>
																						<m:apply id="id82688">
																							<m:minus id="id82689"/>
																							<m:cn id="id82690">1</m:cn>
																						</m:apply>
																					</m:apply>
																					<m:cn id="id82692">0</m:cn>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id82694">
																		<m:apply id="id82696">
																			<m:csymbol id="id82697" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id82701"/>
																			<m:apply id="id82702">
																				<m:list id="id82703"/>
																				<m:ci id="id82704">X</m:ci>
																				<m:divide id="id82707"/>
																				<m:apply id="id82708">
																					<m:divide id="id82709"/>
																					<m:ci id="id82710"/>
																					<m:ci id="id82711">T</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id82713">
																			<m:ci id="id82714">~</m:ci>
																			<m:ci id="id82716">a</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:ci id="id82718">⌣</m:ci>
																<m:ci id="id82721">e</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id82724" class="eqpad"/>
										</tr>

									</table>
									<p id="id82730" style="" class="p">
										<span id="id82736" style="" class="text italic">where <m:math id="id82742" display="inline">
												<m:semantics id="id82746">
													<m:mrow id="id82747">
														<m:msub id="id82748">
															<m:mi id="id82749">o</m:mi>
															<m:mi id="id82751">G</m:mi>
														</m:msub>
														<m:mo id="id82753" mathvariant="italic">⁢</m:mo>
														<m:mfenced id="id82758" open="(" close=")">
															<m:mi id="id82763">Y</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:annotation-xml id="id82765" encoding="MathML-Content">
														<m:apply id="id82768">
															<m:times id="id82769"/>
															<m:apply id="id82770">
																<m:csymbol id="id82771" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id82776">o</m:ci>
																<m:ci id="id82778">G</m:ci>
															</m:apply>
															<m:ci id="id82780">Y</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> denotes the order of the maximal subgroup of <m:math id="id82784" display="inline">
												<m:semantics id="id82787">
													<m:mi id="id82788">G</m:mi>
													<m:annotation-xml id="id82791" encoding="MathML-Content">
														<m:ci id="id82794">G</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> which
fixes every point in <m:math id="id82798" display="inline">
												<m:semantics id="id82801">
													<m:mi id="id82802">Y</m:mi>
													<m:annotation-xml id="id82804" encoding="MathML-Content">
														<m:ci id="id82807">Y</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>, and <m:math id="id82811" display="inline">
												<m:semantics id="id82814">
													<m:mfenced id="id82815" open="|" close="|">
														<m:mi id="id82820">W</m:mi>
													</m:mfenced>
													<m:annotation-xml id="id82822" encoding="MathML-Content">
														<m:apply id="id82826">
															<m:abs id="id82827"/>
															<m:ci id="id82828">W</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id82831" display="inline">
												<m:semantics id="id82835">
													<m:mi id="id82836">e</m:mi>
													<m:annotation-xml id="id82838" encoding="MathML-Content">
														<m:ci id="id82841">e</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> are as defined in
theorem B.</span>
									</p>
								</div>

							</div>

						</div>

						<div class="subsection" id="S6.SS2">
							<h3 id="id82856" class="title subsection-title">
								<span id="id82859" class="tag">6.2 </span>The case in which <m:math id="id82865" display="inline">
									<m:semantics id="id82868">
										<m:msub id="id82870">
											<m:mi id="id82871">μ</m:mi>
											<m:mi id="id82873">T</m:mi>
										</m:msub>
										<m:annotation-xml id="id82875" encoding="MathML-Content">
											<m:apply id="id82878">
												<m:csymbol id="id82880" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id82884">μ</m:ci>
												<m:ci id="id82887">T</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is proper, but does not have <m:math id="id82890" display="inline">
									<m:semantics id="id82894">
										<m:mn id="id82895">0</m:mn>
										<m:annotation-xml id="id82897" encoding="MathML-Content">
											<m:cn id="id82900">0</m:cn>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
as a regular value</h3>
							<div class="para" id="S6.SS2.p1">
								<p id="id82911" style="" class="p">The integration formula, theorem B<m:math id="id82918" display="inline">
										<m:semantics id="id82921">
											<m:msup id="id82922">
												<m:mi id="id82924"/>
												<m:mo id="id82925">′</m:mo>
											</m:msup>
											<m:annotation-xml id="id82927" encoding="MathML-Content">
												<m:apply id="id82930">
													<m:ci id="id82931">′</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, can be generalized to the
case in which <m:math id="id82935" display="inline">
										<m:semantics id="id82939">
											<m:mn id="id82940">0</m:mn>
											<m:annotation-xml id="id82942" encoding="MathML-Content">
												<m:cn id="id82945">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is not a regular value for <m:math id="id82949" display="inline">
										<m:semantics id="id82952">
											<m:msub id="id82953">
												<m:mi id="id82954">μ</m:mi>
												<m:mi id="id82957">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id82959" encoding="MathML-Content">
												<m:apply id="id82962">
													<m:csymbol id="id82963" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id82968">μ</m:ci>
													<m:ci id="id82970">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in two different
ways. One way involves perturbing the value at which we take the
symplectic quotient by <m:math id="id82975" display="inline">
										<m:semantics id="id82978">
											<m:mi id="id82979">T</m:mi>
											<m:annotation-xml id="id82981" encoding="MathML-Content">
												<m:ci id="id82984">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, which we now describe. We will then describe
the other alternative, which makes use of compactly-supported
cohomology: that alternative can also handle the case in which <m:math id="id82989" display="inline">
										<m:semantics id="id82993">
											<m:msub id="id82994">
												<m:mi id="id82995">μ</m:mi>
												<m:mi id="id82997">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id82999" encoding="MathML-Content">
												<m:apply id="id83003">
													<m:csymbol id="id83004" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id83008">μ</m:ci>
													<m:ci id="id83011">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
is not compact.</p>
							</div>

							<div class="para" id="S6.SS2.p2">
								<p id="id83023" style="" class="p">A tubular neighbourhood of <m:math id="id83030" display="inline">
										<m:semantics id="id83033">
											<m:mrow id="id83034">
												<m:mrow id="id83035">
													<m:msubsup id="id83036">
														<m:mi id="id83037">μ</m:mi>
														<m:mi id="id83040">G</m:mi>
														<m:mrow id="id83042">
															<m:mo id="id83043">-</m:mo>
															<m:mn id="id83045">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id83047">⁢</m:mo>
													<m:mfenced id="id83049" open="(" close=")">
														<m:mn id="id83054">0</m:mn>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id83057">/</m:mo>
												<m:mi id="id83059">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id83061" encoding="MathML-Content">
												<m:apply id="id83064">
													<m:divide id="id83065"/>
													<m:apply id="id83066">
														<m:times id="id83067"/>
														<m:apply id="id83068">
															<m:csymbol id="id83070" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id83074">
																<m:csymbol id="id83075" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id83080">μ</m:ci>
																<m:ci id="id83082">G</m:ci>
															</m:apply>
															<m:apply id="id83084">
																<m:minus id="id83086"/>
																<m:cn id="id83087">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id83089">0</m:cn>
													</m:apply>
													<m:ci id="id83091">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is an orbifold, since we have
assumed that <m:math id="id83095" display="inline">
										<m:semantics id="id83098">
											<m:mn id="id83099">0</m:mn>
											<m:annotation-xml id="id83101" encoding="MathML-Content">
												<m:cn id="id83104">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a regular value for <m:math id="id83108" display="inline">
										<m:semantics id="id83111">
											<m:msub id="id83112">
												<m:mi id="id83114">μ</m:mi>
												<m:mi id="id83116">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id83118" encoding="MathML-Content">
												<m:apply id="id83121">
													<m:csymbol id="id83122" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id83127">μ</m:ci>
													<m:ci id="id83130">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.  By assumption <m:math id="id83133" display="inline">
										<m:semantics id="id83136">
											<m:mn id="id83137">0</m:mn>
											<m:annotation-xml id="id83140" encoding="MathML-Content">
												<m:cn id="id83143">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is
not a regular value for <m:math id="id83146" display="inline">
										<m:semantics id="id83150">
											<m:msub id="id83151">
												<m:mi id="id83152">μ</m:mi>
												<m:mi id="id83154">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id83156" encoding="MathML-Content">
												<m:apply id="id83160">
													<m:csymbol id="id83161" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id83166">μ</m:ci>
													<m:ci id="id83168">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, but by transversality there exist
regular values arbitrarily close to <m:math id="id83172" display="inline">
										<m:semantics id="id83175">
											<m:mn id="id83176">0</m:mn>
											<m:annotation-xml id="id83178" encoding="MathML-Content">
												<m:cn id="id83182">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.  Let <m:math id="id83185" display="inline">
										<m:semantics id="id83189">
											<m:mrow id="id83190">
												<m:mi id="id83191">ϵ</m:mi>
												<m:mo id="id83193">∈</m:mo>
												<m:msup id="id83196">
													<m:mi id="id83197" mathvariant="fraktur">t</m:mi>
													<m:mo id="id83201">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id83203" encoding="MathML-Content">
												<m:apply id="id83206">
													<m:ci id="id83208">∈</m:ci>
													<m:ci id="id83210">ϵ</m:ci>
													<m:apply id="id83212">
														<m:csymbol id="id83213" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id83218">t</m:ci>
														<m:times id="id83220"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> be a
regular value, and consider the family of symplectic quotients
									<m:math id="id83223" display="inline">
										<m:semantics id="id83226">
											<m:mrow id="id83227">
												<m:mi id="id83228">X</m:mi>
												<m:mo id="id83231">/</m:mo>
												<m:mo id="id83233">/</m:mo>
												<m:mi id="id83235">T</m:mi>
												<m:mo id="id83237">(</m:mo>
												<m:mi id="id83239">p</m:mi>
												<m:mo id="id83241">)</m:mo>
												<m:mo id="id83243">:=</m:mo>
												<m:msubsup id="id83246">
													<m:mi id="id83247">μ</m:mi>
													<m:mi id="id83249">T</m:mi>
													<m:mrow id="id83251">
														<m:mo id="id83252">-</m:mo>
														<m:mn id="id83254">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id83256">(</m:mo>
												<m:mi id="id83259">p</m:mi>
												<m:mo id="id83261">)</m:mo>
												<m:mo id="id83263">/</m:mo>
												<m:mi id="id83265">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id83267" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>, as <m:math id="id83272" display="inline">
										<m:semantics id="id83275">
											<m:mi id="id83276">p</m:mi>
											<m:annotation-xml id="id83278" encoding="MathML-Content">
												<m:ci id="id83282">p</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> moves between <m:math id="id83285" display="inline">
										<m:semantics id="id83288">
											<m:mn id="id83290">0</m:mn>
											<m:annotation-xml id="id83292" encoding="MathML-Content">
												<m:cn id="id83295">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id83298" display="inline">
										<m:semantics id="id83302">
											<m:mi id="id83303">ϵ</m:mi>
											<m:annotation-xml id="id83305" encoding="MathML-Content">
												<m:ci id="id83309">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.  If
									<m:math id="id83312" display="inline">
										<m:semantics id="id83316">
											<m:mi id="id83317">ϵ</m:mi>
											<m:annotation-xml id="id83319" encoding="MathML-Content">
												<m:ci id="id83322">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is sufficiently close to <m:math id="id83326" display="inline">
										<m:semantics id="id83330">
											<m:mn id="id83331">0</m:mn>
											<m:annotation-xml id="id83333" encoding="MathML-Content">
												<m:cn id="id83336">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, then we can find diffeomorphisms
between a neighbourhood of <m:math id="id83340" display="inline">
										<m:semantics id="id83343">
											<m:mrow id="id83344">
												<m:mrow id="id83346">
													<m:msubsup id="id83347">
														<m:mi id="id83348">μ</m:mi>
														<m:mi id="id83350">G</m:mi>
														<m:mrow id="id83352">
															<m:mo id="id83353">-</m:mo>
															<m:mn id="id83355">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id83358">⁢</m:mo>
													<m:mfenced id="id83360" open="(" close=")">
														<m:mn id="id83365">0</m:mn>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id83367">/</m:mo>
												<m:mi id="id83369">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id83371" encoding="MathML-Content">
												<m:apply id="id83375">
													<m:divide id="id83376"/>
													<m:apply id="id83377">
														<m:times id="id83378"/>
														<m:apply id="id83379">
															<m:csymbol id="id83380" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id83385">
																<m:csymbol id="id83386" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id83390">μ</m:ci>
																<m:ci id="id83393">G</m:ci>
															</m:apply>
															<m:apply id="id83395">
																<m:minus id="id83396"/>
																<m:cn id="id83397">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id83399">0</m:cn>
													</m:apply>
													<m:ci id="id83401">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and open sets in the quotients
									<m:math id="id83405" display="inline">
										<m:semantics id="id83408">
											<m:mrow id="id83409">
												<m:mi id="id83410">X</m:mi>
												<m:mo id="id83413">/</m:mo>
												<m:mo id="id83415">/</m:mo>
												<m:mi id="id83417">T</m:mi>
												<m:mo id="id83419">(</m:mo>
												<m:mi id="id83421">p</m:mi>
												<m:mo id="id83423">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83425" encoding="MathML-Content">
												<m:ci id="id83429">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. To do this, we note that the (orbifold) vector bundle <m:math id="id83433" display="inline">
										<m:semantics id="id83436">
											<m:mrow id="id83437">
												<m:msub id="id83438">
													<m:mo id="id83439">⊕</m:mo>
													<m:mrow id="id83442">
														<m:mi id="id83443">α</m:mi>
														<m:mo id="id83445">∈</m:mo>
														<m:msup id="id83447">
															<m:mi id="id83448" mathvariant="normal">Δ</m:mi>
															<m:mo id="id83453">-</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:msub id="id83455">
													<m:mi id="id83456">L</m:mi>
													<m:mi id="id83458">α</m:mi>
												</m:msub>
												<m:mo id="id83461">→</m:mo>
												<m:mi id="id83463">X</m:mi>
												<m:mo id="id83465">/</m:mo>
												<m:mo id="id83468">/</m:mo>
												<m:mi id="id83470">T</m:mi>
												<m:mo id="id83472">(</m:mo>
												<m:mn id="id83474">0</m:mn>
												<m:mo id="id83476">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83478" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> with section <m:math id="id83483" display="inline">
										<m:semantics id="id83486">
											<m:mi id="id83487">s</m:mi>
											<m:annotation-xml id="id83489" encoding="MathML-Content">
												<m:ci id="id83493">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, defined in proposition <a id="id83498" href="#S1.E2" title="Proposition 1.2 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id83496" style="" class="text tag">1.2</span>
									</a>, is
naturally defined over all quotients <m:math id="id83513" display="inline">
										<m:semantics id="id83516">
											<m:mrow id="id83517">
												<m:mi id="id83518">X</m:mi>
												<m:mo id="id83521">/</m:mo>
												<m:mo id="id83523">/</m:mo>
												<m:mi id="id83525">T</m:mi>
												<m:mo id="id83527">(</m:mo>
												<m:mi id="id83529">p</m:mi>
												<m:mo id="id83531">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83533" encoding="MathML-Content">
												<m:ci id="id83537">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. For <m:math id="id83540" display="inline">
										<m:semantics id="id83544">
											<m:mi id="id83545">ϵ</m:mi>
											<m:annotation-xml id="id83547" encoding="MathML-Content">
												<m:ci id="id83550">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
sufficiently small, the section <m:math id="id83554" display="inline">
										<m:semantics id="id83558">
											<m:mi id="id83559">s</m:mi>
											<m:annotation-xml id="id83561" encoding="MathML-Content">
												<m:ci id="id83564">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> will remain transverse to the
zero-section, and hence the zeros of <m:math id="id83568" display="inline">
										<m:semantics id="id83571">
											<m:mi id="id83572">s</m:mi>
											<m:annotation-xml id="id83574" encoding="MathML-Content">
												<m:ci id="id83578">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> on the symplectic quotients
									<m:math id="id83581" display="inline">
										<m:semantics id="id83585">
											<m:mrow id="id83586">
												<m:mi id="id83587">X</m:mi>
												<m:mo id="id83589">/</m:mo>
												<m:mo id="id83591">/</m:mo>
												<m:mi id="id83593">T</m:mi>
												<m:mo id="id83595">(</m:mo>
												<m:mi id="id83598">p</m:mi>
												<m:mo id="id83600">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83602" encoding="MathML-Content">
												<m:ci id="id83605">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> will be diffeomorphic to one another, along with tubular
neighbourhoods of these zerosets. Thus, for <m:math id="id83609" display="inline">
										<m:semantics id="id83613">
											<m:mi id="id83614">ϵ</m:mi>
											<m:annotation-xml id="id83616" encoding="MathML-Content">
												<m:ci id="id83619">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> sufficiently small,
we have an injection <m:math id="id83623" display="inline">
										<m:semantics id="id83627">
											<m:mrow id="id83628">
												<m:msup id="id83629">
													<m:mi id="id83630">i</m:mi>
													<m:mo id="id83632">′</m:mo>
												</m:msup>
												<m:mo id="id83634">:</m:mo>
												<m:msubsup id="id83637">
													<m:mi id="id83638">μ</m:mi>
													<m:mi id="id83640">G</m:mi>
													<m:mrow id="id83642">
														<m:mo id="id83643">-</m:mo>
														<m:mn id="id83645">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id83648">(</m:mo>
												<m:mn id="id83650">0</m:mn>
												<m:mo id="id83652">)</m:mo>
												<m:mo id="id83654">/</m:mo>
												<m:mi id="id83656">T</m:mi>
												<m:mo id="id83658">↪</m:mo>
												<m:mi id="id83661">X</m:mi>
												<m:mo id="id83663">/</m:mo>
												<m:mo id="id83665">/</m:mo>
												<m:mi id="id83667">T</m:mi>
												<m:mo id="id83669">(</m:mo>
												<m:mi id="id83671">ϵ</m:mi>
												<m:mo id="id83674">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83676" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>, and the
main topological result, proposition <a id="id83683" href="#S1.E2" title="Proposition 1.2 ‣ 1 The main topological result ‣ Symplectic quotients by a nonabelian groupand by its maximal torus" class="ref">
										<span id="id83681" style="" class="text tag">1.2</span>
									</a> applies with the
map <m:math id="id83698" display="inline">
										<m:semantics id="id83701">
											<m:msup id="id83702">
												<m:mi id="id83703">i</m:mi>
												<m:mo id="id83705">′</m:mo>
											</m:msup>
											<m:annotation-xml id="id83707" encoding="MathML-Content">
												<m:apply id="id83711">
													<m:csymbol id="id83712" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id83716">i</m:ci>
													<m:ci id="id83719">′</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in place of the map <m:math id="id83722" display="inline">
										<m:semantics id="id83726">
											<m:mi id="id83727">i</m:mi>
											<m:annotation-xml id="id83729" encoding="MathML-Content">
												<m:ci id="id83732">i</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
A sufficient condition on <m:math id="id83736" display="inline">
										<m:semantics id="id83739">
											<m:mi id="id83740">ϵ</m:mi>
											<m:annotation-xml id="id83743" encoding="MathML-Content">
												<m:ci id="id83746">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is that there exist path joining <m:math id="id83750" display="inline">
										<m:semantics id="id83753">
											<m:mi id="id83754">ϵ</m:mi>
											<m:annotation-xml id="id83757" encoding="MathML-Content">
												<m:ci id="id83760">ϵ</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
and <m:math id="id83764" display="inline">
										<m:semantics id="id83767">
											<m:mn id="id83768">0</m:mn>
											<m:annotation-xml id="id83770" encoding="MathML-Content">
												<m:cn id="id83774">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, consisting entirely of regular values (except of course for
									<m:math id="id83777" display="inline">
										<m:semantics id="id83781">
											<m:mn id="id83782">0</m:mn>
											<m:annotation-xml id="id83784" encoding="MathML-Content">
												<m:cn id="id83787">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math>). Note also that the notion of a
‘lift’ of a cohomology class from <m:math id="id83792" display="inline">
										<m:semantics id="id83795">
											<m:mrow id="id83796">
												<m:mi id="id83797">X</m:mi>
												<m:mo id="id83799">/</m:mo>
												<m:mo id="id83802">/</m:mo>
												<m:mi id="id83804">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id83806" encoding="MathML-Content">
												<m:ci id="id83809">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> to <m:math id="id83813" display="inline">
										<m:semantics id="id83816">
											<m:mrow id="id83817">
												<m:mi id="id83818">X</m:mi>
												<m:mo id="id83820">/</m:mo>
												<m:mo id="id83822">/</m:mo>
												<m:mi id="id83824">T</m:mi>
												<m:mo id="id83827">(</m:mo>
												<m:mi id="id83829">ϵ</m:mi>
												<m:mo id="id83831">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id83833" encoding="MathML-Content">
												<m:ci id="id83837">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is
well-defined in this case.</p>
							</div>

							<div class="theorem theoremBpp" id="ThmtheoremBppx1">
								<h6 id="id83849" class="title theorem-title bold">
									<span id="id83852" class="tag">Theorem B<m:math id="id83857" display="inline">
											<m:semantics id="id83860">
												<m:mmultiscripts id="id83861">
													<m:msup id="id83862">
														<m:mi id="id83863"/>
														<m:mo id="id83864" mathvariant="normal">′</m:mo>
													</m:msup>
													<m:mprescripts id="id83869"/>
													<m:none id="id83870"/>
													<m:mo id="id83871" mathvariant="normal">′</m:mo>
												</m:mmultiscripts>
												<m:annotation-xml id="id83876" encoding="MathML-Content">
													<m:apply id="id83879">
														<m:csymbol id="id83880" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id83885">
															<m:ci id="id83886">′</m:ci>
														</m:apply>
														<m:ci id="id83888">′</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</span> (Integration formula).</h6>
								<div class="para" id="ThmtheoremBppx1.p1">
									<p id="id83900" style="" class="p">
										<span id="id83905" style="" class="text italic">Suppose <m:math id="id83912" display="inline">
												<m:semantics id="id83915">
													<m:msub id="id83916">
														<m:mi id="id83917">μ</m:mi>
														<m:mi id="id83920">G</m:mi>
													</m:msub>
													<m:annotation-xml id="id83922" encoding="MathML-Content">
														<m:apply id="id83925">
															<m:csymbol id="id83926" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id83931">μ</m:ci>
															<m:ci id="id83933">G</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id83937" display="inline">
												<m:semantics id="id83940">
													<m:msub id="id83941">
														<m:mi id="id83942">μ</m:mi>
														<m:mi id="id83944">T</m:mi>
													</m:msub>
													<m:annotation-xml id="id83947" encoding="MathML-Content">
														<m:apply id="id83950">
															<m:csymbol id="id83951" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id83956">μ</m:ci>
															<m:ci id="id83958">T</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> are proper maps, and <m:math id="id83962" display="inline">
												<m:semantics id="id83965">
													<m:mn id="id83966" mathvariant="normal">0</m:mn>
													<m:annotation-xml id="id83970" encoding="MathML-Content">
														<m:cn id="id83974">0</m:cn>
													</m:annotation-xml>
												</m:semantics>
											</m:math> is a regular value for
											<m:math id="id83977" display="inline">
												<m:semantics id="id83981">
													<m:msub id="id83982">
														<m:mi id="id83983">μ</m:mi>
														<m:mi id="id83985">G</m:mi>
													</m:msub>
													<m:annotation-xml id="id83987" encoding="MathML-Content">
														<m:apply id="id83991">
															<m:csymbol id="id83992" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id83996">μ</m:ci>
															<m:ci id="id83999">G</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>. Then for any regular value
											<m:math id="id84002" display="inline">
												<m:semantics id="id84006">
													<m:mrow id="id84007">
														<m:mi id="id84008">ϵ</m:mi>
														<m:mo id="id84010" mathvariant="normal">∈</m:mo>
														<m:msup id="id84015">
															<m:mi id="id84016" mathvariant="fraktur">t</m:mi>
															<m:mo id="id84020" mathvariant="normal">*</m:mo>
														</m:msup>
													</m:mrow>
													<m:annotation-xml id="id84025" encoding="MathML-Content">
														<m:apply id="id84028">
															<m:ci id="id84029">∈</m:ci>
															<m:ci id="id84032">ϵ</m:ci>
															<m:apply id="id84034">
																<m:csymbol id="id84035" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id84040">t</m:ci>
																<m:times id="id84042"/>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> sufficiently close to <m:math id="id84044" display="inline">
												<m:semantics id="id84048">
													<m:mrow id="id84049">
														<m:mn id="id84050" mathvariant="normal">0</m:mn>
														<m:mo id="id84054" mathvariant="normal">∈</m:mo>
														<m:msup id="id84059">
															<m:mi id="id84060" mathvariant="fraktur">t</m:mi>
															<m:mo id="id84064" mathvariant="normal">*</m:mo>
														</m:msup>
													</m:mrow>
													<m:annotation-xml id="id84069" encoding="MathML-Content">
														<m:apply id="id84072">
															<m:ci id="id84073">∈</m:ci>
															<m:cn id="id84075">0</m:cn>
															<m:apply id="id84078">
																<m:csymbol id="id84079" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id84083">t</m:ci>
																<m:times id="id84085"/>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>, and
any class <m:math id="id84088" display="inline">
												<m:semantics id="id84091">
													<m:mrow id="id84092">
														<m:mi id="id84093">a</m:mi>
														<m:mo id="id84095" mathvariant="normal">∈</m:mo>
														<m:msup id="id84100">
															<m:mi id="id84101">H</m:mi>
															<m:mo id="id84103" mathvariant="normal">*</m:mo>
														</m:msup>
														<m:mo id="id84108" mathvariant="normal">(</m:mo>
														<m:mi id="id84112">X</m:mi>
														<m:mo id="id84114" mathvariant="normal">/</m:mo>
														<m:mo id="id84119" mathvariant="normal">/</m:mo>
														<m:mi id="id84123">G</m:mi>
														<m:mo id="id84125" mathvariant="normal">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id84130" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>,  with lift <m:math id="id84134" display="inline">
												<m:semantics id="id84138">
													<m:mrow id="id84139">
														<m:mover id="id84140" accent="true">
															<m:mi id="id84143">a</m:mi>
															<m:mo id="id84145" mathvariant="normal">~</m:mo>
														</m:mover>
														<m:mo id="id84150" mathvariant="normal">∈</m:mo>
														<m:msup id="id84154">
															<m:mi id="id84155">H</m:mi>
															<m:mo id="id84157" mathvariant="normal">*</m:mo>
														</m:msup>
														<m:mo id="id84162" mathvariant="normal">(</m:mo>
														<m:mi id="id84166">X</m:mi>
														<m:mo id="id84168" mathvariant="normal">/</m:mo>
														<m:mo id="id84173" mathvariant="normal">/</m:mo>
														<m:mi id="id84177">T</m:mi>
														<m:mo id="id84179" mathvariant="normal">(</m:mo>
														<m:mi id="id84184">ϵ</m:mi>
														<m:mo id="id84186" mathvariant="normal">)</m:mo>
														<m:mo id="id84190" mathvariant="normal">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id84195" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>,</span>
									</p>
									<table class="equation" id="S6.Ex2">

										<tr valign="baseline" class="equation" id="S6.Ex2">
											<td id="id84216" class="eqpad"/>
											<td id="id84220" nowrap="yes" align="center" colspan="1">
												<m:math id="id84228" display="block">
													<m:semantics id="id84231">
														<m:mrow id="id84232">
															<m:mrow id="id84233">
																<m:mrow id="id84234">
																	<m:msub id="id84235">
																		<m:mo id="id84236">∫</m:mo>
																		<m:mrow id="id84239">
																			<m:mi id="id84240">X</m:mi>
																			<m:mo id="id84242">⁣</m:mo>
																			<m:mo id="id84244">/</m:mo>
																			<m:mo id="id84246">⁣</m:mo>
																			<m:mrow id="id84249">
																				<m:none id="id84250"/>
																				<m:mo id="id84251">/</m:mo>
																				<m:mi id="id84253">G</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msub>
																	<m:mi id="id84255">a</m:mi>
																</m:mrow>
																<m:mo id="id84257">=</m:mo>
																<m:mrow id="id84260">
																	<m:mrow id="id84261">
																		<m:mfrac id="id84262">
																			<m:mn id="id84263">1</m:mn>
																			<m:mfenced id="id84265" open="|" close="|">
																				<m:mi id="id84270">W</m:mi>
																			</m:mfenced>
																		</m:mfrac>
																		<m:mo id="id84272">⋅</m:mo>
																		<m:mfrac id="id84274">
																			<m:mrow id="id84276">
																				<m:msub id="id84277">
																					<m:mi id="id84278">o</m:mi>
																					<m:mi id="id84280">T</m:mi>
																				</m:msub>
																				<m:mo id="id84282">⁢</m:mo>
																				<m:mfenced id="id84284" open="(" close=")">
																					<m:mrow id="id84289">
																						<m:msubsup id="id84290">
																							<m:mi id="id84292">μ</m:mi>
																							<m:mi id="id84294">T</m:mi>
																							<m:mrow id="id84296">
																								<m:mo id="id84297">-</m:mo>
																								<m:mn id="id84299">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id84301">⁢</m:mo>
																						<m:mfenced id="id84304" open="(" close=")">
																							<m:mn id="id84309">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																			<m:mrow id="id84311">
																				<m:msub id="id84312">
																					<m:mi id="id84313">o</m:mi>
																					<m:mi id="id84315">G</m:mi>
																				</m:msub>
																				<m:mo id="id84317">⁢</m:mo>
																				<m:mfenced id="id84320" open="(" close=")">
																					<m:mrow id="id84325">
																						<m:msubsup id="id84326">
																							<m:mi id="id84327">μ</m:mi>
																							<m:mi id="id84329">G</m:mi>
																							<m:mrow id="id84332">
																								<m:mo id="id84333">-</m:mo>
																								<m:mn id="id84335">1</m:mn>
																							</m:mrow>
																						</m:msubsup>
																						<m:mo id="id84337">⁢</m:mo>
																						<m:mfenced id="id84339" open="(" close=")">
																							<m:mn id="id84344">0</m:mn>
																						</m:mfenced>
																					</m:mrow>
																				</m:mfenced>
																			</m:mrow>
																		</m:mfrac>
																	</m:mrow>
																	<m:mo id="id84346">⁢</m:mo>
																	<m:mrow id="id84349">
																		<m:msub id="id84350">
																			<m:mo id="id84351">∫</m:mo>
																			<m:mrow id="id84353">
																				<m:mi id="id84354">X</m:mi>
																				<m:mo id="id84357">⁣</m:mo>
																				<m:mo id="id84359">/</m:mo>
																				<m:mo id="id84361">⁣</m:mo>
																				<m:mrow id="id84364">
																					<m:none id="id84365"/>
																					<m:mo id="id84366">/</m:mo>
																					<m:mrow id="id84368">
																						<m:mi id="id84369">T</m:mi>
																						<m:mo id="id84371">⁢</m:mo>
																						<m:mfenced id="id84373" open="(" close=")">
																							<m:mi id="id84378">ϵ</m:mi>
																						</m:mfenced>
																					</m:mrow>
																				</m:mrow>
																			</m:mrow>
																		</m:msub>
																		<m:mover id="id84381" accent="true">
																			<m:mi id="id84384">a</m:mi>
																			<m:mo id="id84386">~</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mrow>
																<m:mo id="id84388">⌣</m:mo>
																<m:mi id="id84391">e</m:mi>
															</m:mrow>
															<m:mo id="id84393">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id84395" encoding="MathML-Content">
															<m:apply id="id84398">
																<m:ci id="id84400"/>
																<m:apply id="id84401">
																	<m:apply id="id84402">
																		<m:csymbol id="id84403" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id84407"/>
																		<m:apply id="id84408">
																			<m:list id="id84410"/>
																			<m:ci id="id84411">X</m:ci>
																			<m:divide id="id84413"/>
																			<m:apply id="id84414">
																				<m:divide id="id84415"/>
																				<m:ci id="id84416"/>
																				<m:ci id="id84417">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci id="id84419">a</m:ci>
																</m:apply>
																<m:eq id="id84421"/>
																<m:apply id="id84422">
																	<m:times id="id84423"/>
																	<m:apply id="id84424">
																		<m:ci id="id84426">⋅</m:ci>
																		<m:apply id="id84428">
																			<m:divide id="id84429"/>
																			<m:cn id="id84430">1</m:cn>
																			<m:apply id="id84432">
																				<m:abs id="id84433"/>
																				<m:ci id="id84434">W</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id84436">
																			<m:divide id="id84438"/>
																			<m:apply id="id84439">
																				<m:times id="id84440"/>
																				<m:apply id="id84441">
																					<m:csymbol id="id84442" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id84446">o</m:ci>
																					<m:ci id="id84449">T</m:ci>
																				</m:apply>
																				<m:apply id="id84451">
																					<m:times id="id84452"/>
																					<m:apply id="id84453">
																						<m:csymbol id="id84454" cd="ambiguous">superscript</m:csymbol>
																						<m:apply id="id84459">
																							<m:csymbol id="id84460" cd="ambiguous">subscript</m:csymbol>
																							<m:ci id="id84464">μ</m:ci>
																							<m:ci id="id84467">T</m:ci>
																						</m:apply>
																						<m:apply id="id84469">
																							<m:minus id="id84470"/>
																							<m:cn id="id84471">1</m:cn>
																						</m:apply>
																					</m:apply>
																					<m:cn id="id84473">0</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id84475">
																				<m:times id="id84476"/>
																				<m:apply id="id84477">
																					<m:csymbol id="id84478" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id84483">o</m:ci>
																					<m:ci id="id84485">G</m:ci>
																				</m:apply>
																				<m:apply id="id84487">
																					<m:times id="id84488"/>
																					<m:apply id="id84490">
																						<m:csymbol id="id84491" cd="ambiguous">superscript</m:csymbol>
																						<m:apply id="id84495">
																							<m:csymbol id="id84496" cd="ambiguous">subscript</m:csymbol>
																							<m:ci id="id84501">μ</m:ci>
																							<m:ci id="id84503">G</m:ci>
																						</m:apply>
																						<m:apply id="id84506">
																							<m:minus id="id84507"/>
																							<m:cn id="id84508">1</m:cn>
																						</m:apply>
																					</m:apply>
																					<m:cn id="id84510">0</m:cn>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id84512">
																		<m:apply id="id84513">
																			<m:csymbol id="id84514" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id84519"/>
																			<m:apply id="id84520">
																				<m:list id="id84521"/>
																				<m:ci id="id84522">X</m:ci>
																				<m:divide id="id84524"/>
																				<m:apply id="id84525">
																					<m:divide id="id84526"/>
																					<m:ci id="id84527"/>
																					<m:apply id="id84528">
																						<m:times id="id84529"/>
																						<m:ci id="id84530">T</m:ci>
																						<m:ci id="id84533">ϵ</m:ci>
																					</m:apply>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id84535">
																			<m:ci id="id84536">~</m:ci>
																			<m:ci id="id84538">a</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:ci id="id84540">⌣</m:ci>
																<m:ci id="id84543">e</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id84546" class="eqpad"/>
										</tr>

									</table>
									<p id="id84552" style="" class="p">
										<span id="id84558" style="" class="text italic">where <m:math id="id84564" display="inline">
												<m:semantics id="id84568">
													<m:mrow id="id84569">
														<m:msub id="id84570">
															<m:mi id="id84571">o</m:mi>
															<m:mi id="id84573">G</m:mi>
														</m:msub>
														<m:mo id="id84575" mathvariant="italic">⁢</m:mo>
														<m:mfenced id="id84580" open="(" close=")">
															<m:mi id="id84585">Y</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:annotation-xml id="id84587" encoding="MathML-Content">
														<m:apply id="id84590">
															<m:times id="id84591"/>
															<m:apply id="id84592">
																<m:csymbol id="id84593" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id84598">o</m:ci>
																<m:ci id="id84600">G</m:ci>
															</m:apply>
															<m:ci id="id84602">Y</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> denotes the order of the maximal subgroup of <m:math id="id84606" display="inline">
												<m:semantics id="id84609">
													<m:mi id="id84610">G</m:mi>
													<m:annotation-xml id="id84613" encoding="MathML-Content">
														<m:ci id="id84616">G</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> which
fixes every point in <m:math id="id84620" display="inline">
												<m:semantics id="id84623">
													<m:mi id="id84624">Y</m:mi>
													<m:annotation-xml id="id84626" encoding="MathML-Content">
														<m:ci id="id84629">Y</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>, and <m:math id="id84633" display="inline">
												<m:semantics id="id84636">
													<m:mfenced id="id84637" open="|" close="|">
														<m:mi id="id84642">W</m:mi>
													</m:mfenced>
													<m:annotation-xml id="id84644" encoding="MathML-Content">
														<m:apply id="id84648">
															<m:abs id="id84649"/>
															<m:ci id="id84650">W</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id84653" display="inline">
												<m:semantics id="id84657">
													<m:mi id="id84658">e</m:mi>
													<m:annotation-xml id="id84660" encoding="MathML-Content">
														<m:ci id="id84663">e</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> are as defined in
theorem B.</span>
									</p>
								</div>

							</div>

						</div>

						<div class="subsection" id="S6.SS3">
							<h3 id="id84678" class="title subsection-title">
								<span id="id84681" class="tag">6.3 </span>The case in which <m:math id="id84687" display="inline">
									<m:semantics id="id84690">
										<m:msub id="id84692">
											<m:mi id="id84693">μ</m:mi>
											<m:mi id="id84695">T</m:mi>
										</m:msub>
										<m:annotation-xml id="id84697" encoding="MathML-Content">
											<m:apply id="id84700">
												<m:csymbol id="id84702" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id84706">μ</m:ci>
												<m:ci id="id84709">T</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is not proper</h3>
							<div class="para" id="S6.SS3.p1">
								<p id="id84720" style="" class="p">If <m:math id="id84726" display="inline">
										<m:semantics id="id84729">
											<m:msub id="id84730">
												<m:mi id="id84731">μ</m:mi>
												<m:mi id="id84734">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id84736" encoding="MathML-Content">
												<m:apply id="id84739">
													<m:csymbol id="id84740" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id84745">μ</m:ci>
													<m:ci id="id84747">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is not proper but <m:math id="id84751" display="inline">
										<m:semantics id="id84754">
											<m:mrow id="id84755">
												<m:mn id="id84756">0</m:mn>
												<m:mo id="id84758">∈</m:mo>
												<m:msup id="id84761">
													<m:mi id="id84762" mathvariant="fraktur">t</m:mi>
													<m:mo id="id84766">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id84768" encoding="MathML-Content">
												<m:apply id="id84772">
													<m:ci id="id84773">∈</m:ci>
													<m:cn id="id84775">0</m:cn>
													<m:apply id="id84777">
														<m:csymbol id="id84778" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id84783">t</m:ci>
														<m:times id="id84785"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a regular value, then <m:math id="id84788" display="inline">
										<m:semantics id="id84791">
											<m:mrow id="id84792">
												<m:mi id="id84793">X</m:mi>
												<m:mo id="id84795">/</m:mo>
												<m:mo id="id84798">/</m:mo>
												<m:mi id="id84800">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id84802" encoding="MathML-Content">
												<m:ci id="id84805">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
is a noncompact orbifold, with (orbifold) line bundles <m:math id="id84809" display="inline">
										<m:semantics id="id84812">
											<m:msub id="id84813">
												<m:mi id="id84814">L</m:mi>
												<m:mi id="id84817">α</m:mi>
											</m:msub>
											<m:annotation-xml id="id84819" encoding="MathML-Content">
												<m:apply id="id84822">
													<m:csymbol id="id84823" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id84828">L</m:ci>
													<m:ci id="id84830">α</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and
with <m:math id="id84834" display="inline">
										<m:semantics id="id84837">
											<m:mrow id="id84838">
												<m:mrow id="id84839">
													<m:msubsup id="id84840">
														<m:mi id="id84842">μ</m:mi>
														<m:mi id="id84844">G</m:mi>
														<m:mrow id="id84846">
															<m:mo id="id84847">-</m:mo>
															<m:mn id="id84849">1</m:mn>
														</m:mrow>
													</m:msubsup>
													<m:mo id="id84851">⁢</m:mo>
													<m:mfenced id="id84854" open="(" close=")">
														<m:mn id="id84859">0</m:mn>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id84861">/</m:mo>
												<m:mi id="id84863">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id84865" encoding="MathML-Content">
												<m:apply id="id84869">
													<m:divide id="id84870"/>
													<m:apply id="id84871">
														<m:times id="id84872"/>
														<m:apply id="id84873">
															<m:csymbol id="id84874" cd="ambiguous">superscript</m:csymbol>
															<m:apply id="id84879">
																<m:csymbol id="id84880" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id84884">μ</m:ci>
																<m:ci id="id84887">G</m:ci>
															</m:apply>
															<m:apply id="id84889">
																<m:minus id="id84890"/>
																<m:cn id="id84891">1</m:cn>
															</m:apply>
														</m:apply>
														<m:cn id="id84893">0</m:cn>
													</m:apply>
													<m:ci id="id84895">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> as a compact suborbifold.</p>
							</div>

							<div class="para" id="S6.SS3.p2">
								<p id="id84908" style="" class="p">The section <m:math id="id84914" display="inline">
										<m:semantics id="id84917">
											<m:mi id="id84918">s</m:mi>
											<m:annotation-xml id="id84921" encoding="MathML-Content">
												<m:ci id="id84924">s</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of the bundle <m:math id="id84927" display="inline">
										<m:semantics id="id84931">
											<m:mrow id="id84932">
												<m:msub id="id84933">
													<m:mo id="id84934">⊕</m:mo>
													<m:mrow id="id84936">
														<m:mi id="id84937">α</m:mi>
														<m:mo id="id84940">∈</m:mo>
														<m:msup id="id84942">
															<m:mi id="id84943" mathvariant="normal">Δ</m:mi>
															<m:mo id="id84948">-</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:msub id="id84950">
													<m:mi id="id84951">L</m:mi>
													<m:mi id="id84953">α</m:mi>
												</m:msub>
												<m:mo id="id84956">→</m:mo>
												<m:mi id="id84958">X</m:mi>
												<m:mo id="id84960">/</m:mo>
												<m:mo id="id84962">/</m:mo>
												<m:mi id="id84964">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id84967" encoding="MathML-Content"/>
										</m:semantics>
									</m:math> has compactly-supported
zeroset, and hence the pair <m:math id="id84972" display="inline">
										<m:semantics id="id84975">
											<m:mfenced id="id84976" open="(" close=")">
												<m:mrow id="id84981">
													<m:mrow id="id84982">
														<m:msub id="id84983">
															<m:mo id="id84984">⊕</m:mo>
															<m:mrow id="id84987">
																<m:mi id="id84988">α</m:mi>
																<m:mo id="id84990">∈</m:mo>
																<m:msup id="id84992">
																	<m:mi id="id84994" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id84998">-</m:mo>
																</m:msup>
															</m:mrow>
														</m:msub>
														<m:msub id="id85000">
															<m:mi id="id85001">L</m:mi>
															<m:mi id="id85004">α</m:mi>
														</m:msub>
													</m:mrow>
													<m:mo id="id85006">,</m:mo>
													<m:mi id="id85008">s</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id85010" encoding="MathML-Content">
												<m:apply id="id85014">
													<m:interval id="id85015" closure="open"/>
													<m:apply id="id85018">
														<m:apply id="id85019">
															<m:csymbol id="id85020" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id85025">⊕</m:ci>
															<m:apply id="id85027">
																<m:ci id="id85028">∈</m:ci>
																<m:ci id="id85031">α</m:ci>
																<m:apply id="id85033">
																	<m:csymbol id="id85034" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id85039">Δ</m:ci>
																	<m:minus id="id85041"/>
																</m:apply>
															</m:apply>
														</m:apply>
														<m:apply id="id85042">
															<m:csymbol id="id85043" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id85048">L</m:ci>
															<m:ci id="id85050">α</m:ci>
														</m:apply>
													</m:apply>
													<m:ci id="id85052">s</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> possesses a <span id="id85056" style="" class="text bold">relative
Euler class<span id="id85063" class="note footnote">
											<sup id="id85066" class="mark">6</sup>
											<div id="id85070" class="note_content">
												<sup id="id85074" class="mark">6</sup>Let <m:math id="id85079" display="inline">
													<m:semantics id="id85083">
														<m:mrow id="id85084">
															<m:mi id="id85085">E</m:mi>
															<m:mo id="id85087" mathvariant="normal">→</m:mo>
															<m:mi id="id85092">Y</m:mi>
														</m:mrow>
														<m:annotation-xml id="id85094" encoding="MathML-Content">
															<m:apply id="id85097">
																<m:ci id="id85098">→</m:ci>
																<m:ci id="id85101">E</m:ci>
																<m:ci id="id85103">Y</m:ci>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> be an oriented vector bundle over a
noncompact manifold <m:math id="id85107" display="inline">
													<m:semantics id="id85110">
														<m:mi id="id85111">Y</m:mi>
														<m:annotation-xml id="id85113" encoding="MathML-Content">
															<m:ci id="id85116">Y</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, and let <m:math id="id85120" display="inline">
													<m:semantics id="id85123">
														<m:mi id="id85124">s</m:mi>
														<m:annotation-xml id="id85126" encoding="MathML-Content">
															<m:ci id="id85130">s</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> be a section whose zeroset is
compact. The bundle <m:math id="id85134" display="inline">
													<m:semantics id="id85137">
														<m:mi id="id85138">E</m:mi>
														<m:annotation-xml id="id85140" encoding="MathML-Content">
															<m:ci id="id85143">E</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math> possesses a Thom class <m:math id="id85147" display="inline">
													<m:semantics id="id85150">
														<m:mi id="id85151" mathvariant="normal">Φ</m:mi>
														<m:annotation-xml id="id85156" encoding="MathML-Content">
															<m:ci id="id85159">Φ</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, and we define
the relative Euler class by <m:math id="id85163" display="inline">
													<m:semantics id="id85167">
														<m:mrow id="id85168">
															<m:mrow id="id85169">
																<m:mi id="id85170">e</m:mi>
																<m:mo id="id85172" mathvariant="bold">⁢</m:mo>
																<m:mfenced id="id85177" open="(" close=")">
																	<m:mrow id="id85182">
																		<m:mi id="id85183">E</m:mi>
																		<m:mo id="id85185" mathvariant="bold">,</m:mo>
																		<m:mi id="id85189">s</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id85192" mathvariant="normal">:=</m:mo>
															<m:mrow id="id85196">
																<m:msup id="id85197">
																	<m:mi id="id85198">s</m:mi>
																	<m:mo id="id85200" mathvariant="normal">*</m:mo>
																</m:msup>
																<m:mo id="id85205" mathvariant="bold">⁢</m:mo>
																<m:mi id="id85209" mathvariant="normal">Φ</m:mi>
															</m:mrow>
															<m:mo id="id85214" mathvariant="normal">∈</m:mo>
															<m:mrow id="id85219">
																<m:msubsup id="id85220">
																	<m:mi id="id85221">H</m:mi>
																	<m:mi id="id85223">c</m:mi>
																	<m:mo id="id85225" mathvariant="normal">*</m:mo>
																</m:msubsup>
																<m:mo id="id85229" mathvariant="bold">⁢</m:mo>
																<m:mfenced id="id85234" open="(" close=")">
																	<m:mi id="id85239">Y</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id85241" encoding="MathML-Content">
															<m:apply id="id85245">
																<m:ci id="id85246"/>
																<m:apply id="id85247">
																	<m:times id="id85248"/>
																	<m:ci id="id85249">e</m:ci>
																	<m:apply id="id85251">
																		<m:interval id="id85252" closure="open"/>
																		<m:ci id="id85255">E</m:ci>
																		<m:ci id="id85258">s</m:ci>
																	</m:apply>
																</m:apply>
																<m:ci id="id85260">:=</m:ci>
																<m:apply id="id85262">
																	<m:times id="id85263"/>
																	<m:apply id="id85264">
																		<m:csymbol id="id85265" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id85270">s</m:ci>
																		<m:times id="id85272"/>
																	</m:apply>
																	<m:ci id="id85273">Φ</m:ci>
																</m:apply>
																<m:ci id="id85275">∈</m:ci>
																<m:apply id="id85278">
																	<m:times id="id85279"/>
																	<m:apply id="id85280">
																		<m:csymbol id="id85281" cd="ambiguous">subscript</m:csymbol>
																		<m:apply id="id85286">
																			<m:csymbol id="id85287" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id85291">H</m:ci>
																			<m:times id="id85293"/>
																		</m:apply>
																		<m:ci id="id85294">c</m:ci>
																	</m:apply>
																	<m:ci id="id85297">Y</m:ci>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>. This
cohomology class is an invariant of the homotopy class of <m:math id="id85300" display="inline">
													<m:semantics id="id85304">
														<m:mi id="id85305">s</m:mi>
														<m:annotation-xml id="id85307" encoding="MathML-Content">
															<m:ci id="id85310">s</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, through
homotopies for which the zeroset remains compact at all times. Given a
section in this homotopy class which is transverse to the zero section
of <m:math id="id85315" display="inline">
													<m:semantics id="id85318">
														<m:mi id="id85319">E</m:mi>
														<m:annotation-xml id="id85322" encoding="MathML-Content">
															<m:ci id="id85325">E</m:ci>
														</m:annotation-xml>
													</m:semantics>
												</m:math>, then <m:math id="id85328" display="inline">
													<m:semantics id="id85332">
														<m:mrow id="id85333">
															<m:mi id="id85334">e</m:mi>
															<m:mo id="id85336" mathvariant="bold">⁢</m:mo>
															<m:mfenced id="id85341" open="(" close=")">
																<m:mrow id="id85346">
																	<m:mi id="id85347">E</m:mi>
																	<m:mo id="id85349" mathvariant="bold">,</m:mo>
																	<m:mi id="id85353">s</m:mi>
																</m:mrow>
															</m:mfenced>
														</m:mrow>
														<m:annotation-xml id="id85355" encoding="MathML-Content">
															<m:apply id="id85359">
																<m:times id="id85360"/>
																<m:ci id="id85361">e</m:ci>
																<m:apply id="id85363">
																	<m:interval id="id85364" closure="open"/>
																	<m:ci id="id85367">E</m:ci>
																	<m:ci id="id85370">s</m:ci>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math> represents the Poincaré dual of the compact
submanifold given by the zeroset (Poincaré duality on a noncompact
manifold is an isomorphism between homology and compactly-supported
cohomology, see for example <cite id="id85376" class="cite">[<a id="id85381" href="#bib.bib4" title="" class="ref">4</a>, propositions 6.24 and
6.25]</cite>).
These statements go over to orbifolds, with rational
cohomology.</div>
										</span>
									</span>
								</p>
								<table class="equation" id="S6.Ex3">

									<tr valign="baseline" class="equation" id="S6.Ex3">
										<td id="id85409" class="eqpad"/>
										<td id="id85413" nowrap="yes" align="center" colspan="1">
											<m:math id="id85421" display="block">
												<m:semantics id="id85424">
													<m:mrow id="id85425">
														<m:msup id="id85426">
															<m:mi id="id85427">e</m:mi>
															<m:mo id="id85429">-</m:mo>
														</m:msup>
														<m:mo id="id85432">:=</m:mo>
														<m:mi id="id85434">e</m:mi>
														<m:mo id="id85436">(</m:mo>
														<m:msub id="id85438">
															<m:mo id="id85439">⊕</m:mo>
															<m:mrow id="id85441">
																<m:mi id="id85442">α</m:mi>
																<m:mo id="id85445">∈</m:mo>
																<m:msup id="id85447">
																	<m:mi id="id85448" mathvariant="normal">Δ</m:mi>
																	<m:mo id="id85453">-</m:mo>
																</m:msup>
															</m:mrow>
														</m:msub>
														<m:msub id="id85455">
															<m:mi id="id85456">L</m:mi>
															<m:mi id="id85458">α</m:mi>
														</m:msub>
														<m:mo id="id85461">,</m:mo>
														<m:mi id="id85463">s</m:mi>
														<m:mo id="id85465">)</m:mo>
														<m:mo id="id85467">∈</m:mo>
														<m:msubsup id="id85470">
															<m:mi id="id85471">H</m:mi>
															<m:mi id="id85473">c</m:mi>
															<m:mo id="id85475">*</m:mo>
														</m:msubsup>
														<m:mo id="id85477">(</m:mo>
														<m:mi id="id85479">X</m:mi>
														<m:mo id="id85481">/</m:mo>
														<m:mo id="id85483">/</m:mo>
														<m:mi id="id85486">T</m:mi>
														<m:mo id="id85488">)</m:mo>
														<m:mo id="id85490">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id85492" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>
										</td>
										<td id="id85496" class="eqpad"/>
									</tr>

								</table>
								<p id="id85503" style="" class="p">lying in the cohomology with compact support of <m:math id="id85510" display="inline">
										<m:semantics id="id85513">
											<m:mrow id="id85514">
												<m:mi id="id85515">X</m:mi>
												<m:mo id="id85517">/</m:mo>
												<m:mo id="id85519">/</m:mo>
												<m:mi id="id85521">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id85524" encoding="MathML-Content">
												<m:ci id="id85527">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
Setting <m:math id="id85530" display="inline">
										<m:semantics id="id85534">
											<m:mrow id="id85535">
												<m:msup id="id85536">
													<m:mi id="id85537">e</m:mi>
													<m:mo id="id85539">+</m:mo>
												</m:msup>
												<m:mo id="id85541">:=</m:mo>
												<m:mrow id="id85543">
													<m:mi id="id85544">e</m:mi>
													<m:mo id="id85546">⁢</m:mo>
													<m:mfenced id="id85549" open="(" close=")">
														<m:mrow id="id85554">
															<m:msub id="id85555">
																<m:mo id="id85556">⊕</m:mo>
																<m:mrow id="id85558">
																	<m:mi id="id85560">α</m:mi>
																	<m:mo id="id85562">∈</m:mo>
																	<m:msup id="id85564">
																		<m:mi id="id85565" mathvariant="normal">Δ</m:mi>
																		<m:mo id="id85570">+</m:mo>
																	</m:msup>
																</m:mrow>
															</m:msub>
															<m:msub id="id85572">
																<m:mi id="id85573">L</m:mi>
																<m:mi id="id85575">α</m:mi>
															</m:msub>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id85578" encoding="MathML-Content">
												<m:apply id="id85581">
													<m:ci id="id85582">:=</m:ci>
													<m:apply id="id85584">
														<m:csymbol id="id85585" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id85590">e</m:ci>
														<m:plus id="id85592"/>
													</m:apply>
													<m:apply id="id85593">
														<m:times id="id85594"/>
														<m:ci id="id85595">e</m:ci>
														<m:apply id="id85598">
															<m:apply id="id85599">
																<m:csymbol id="id85600" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id85604">⊕</m:ci>
																<m:apply id="id85607">
																	<m:ci id="id85608">∈</m:ci>
																	<m:ci id="id85610">α</m:ci>
																	<m:apply id="id85613">
																		<m:csymbol id="id85614" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id85618">Δ</m:ci>
																		<m:plus id="id85621"/>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:apply id="id85622">
																<m:csymbol id="id85623" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id85628">L</m:ci>
																<m:ci id="id85630">α</m:ci>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (the regular Euler class), then the product
									<m:math id="id85634" display="inline">
										<m:semantics id="id85637">
											<m:mrow id="id85638">
												<m:mi id="id85639">e</m:mi>
												<m:mo id="id85641">:=</m:mo>
												<m:msup id="id85643">
													<m:mi id="id85644">e</m:mi>
													<m:mo id="id85647">+</m:mo>
												</m:msup>
												<m:mo id="id85649">⌣</m:mo>
												<m:msup id="id85651">
													<m:mi id="id85652">e</m:mi>
													<m:mo id="id85654">-</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id85656" encoding="MathML-Content">
												<m:apply id="id85660">
													<m:ci id="id85661"/>
													<m:ci id="id85662">e</m:ci>
													<m:ci id="id85664">:=</m:ci>
													<m:apply id="id85666">
														<m:csymbol id="id85667" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id85672">e</m:ci>
														<m:plus id="id85674"/>
													</m:apply>
													<m:ci id="id85675">⌣</m:ci>
													<m:apply id="id85678">
														<m:csymbol id="id85679" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id85683">e</m:ci>
														<m:minus id="id85685"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> lies in <m:math id="id85688" display="inline">
										<m:semantics id="id85691">
											<m:mrow id="id85692">
												<m:msubsup id="id85693">
													<m:mi id="id85694">H</m:mi>
													<m:mi id="id85696">c</m:mi>
													<m:mo id="id85699">*</m:mo>
												</m:msubsup>
												<m:mo id="id85701">(</m:mo>
												<m:mi id="id85703">X</m:mi>
												<m:mo id="id85705">/</m:mo>
												<m:mo id="id85707">/</m:mo>
												<m:mi id="id85709">T</m:mi>
												<m:mo id="id85711">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id85714" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>, hence for any class
									<m:math id="id85718" display="inline">
										<m:semantics id="id85722">
											<m:mrow id="id85723">
												<m:mover id="id85724" accent="true">
													<m:mi id="id85727">a</m:mi>
													<m:mo id="id85729">~</m:mo>
												</m:mover>
												<m:mo id="id85731">∈</m:mo>
												<m:msup id="id85734">
													<m:mi id="id85735">H</m:mi>
													<m:mo id="id85737">*</m:mo>
												</m:msup>
												<m:mo id="id85739">(</m:mo>
												<m:mi id="id85741">X</m:mi>
												<m:mo id="id85743">/</m:mo>
												<m:mo id="id85745">/</m:mo>
												<m:mi id="id85748">T</m:mi>
												<m:mo id="id85750">)</m:mo>
											</m:mrow>
											<m:annotation-xml id="id85752" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>, the product <m:math id="id85756" display="inline">
										<m:semantics id="id85760">
											<m:mrow id="id85761">
												<m:mi id="id85762">a</m:mi>
												<m:mo id="id85764">⌣</m:mo>
												<m:mi id="id85766">e</m:mi>
											</m:mrow>
											<m:annotation-xml id="id85769" encoding="MathML-Content">
												<m:apply id="id85772">
													<m:ci id="id85773">⌣</m:ci>
													<m:ci id="id85775">a</m:ci>
													<m:ci id="id85778">e</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> has compact support
and thus has a well-defined integral over
									<m:math id="id85781" display="inline">
										<m:semantics id="id85785">
											<m:mrow id="id85786">
												<m:mi id="id85787">X</m:mi>
												<m:mo id="id85789">/</m:mo>
												<m:mo id="id85791">/</m:mo>
												<m:mi id="id85793">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id85795" encoding="MathML-Content">
												<m:ci id="id85799">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S6.SS3.p3">
								<p id="id85811" style="" class="p">With this interpretation of the class <m:math id="id85817" display="inline">
										<m:semantics id="id85821">
											<m:mi id="id85822">e</m:mi>
											<m:annotation-xml id="id85824" encoding="MathML-Content">
												<m:ci id="id85827">e</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, the integration formula of
theorem B<m:math id="id85831" display="inline">
										<m:semantics id="id85834">
											<m:msup id="id85836">
												<m:mi id="id85837"/>
												<m:mo id="id85838">′</m:mo>
											</m:msup>
											<m:annotation-xml id="id85840" encoding="MathML-Content">
												<m:apply id="id85843">
													<m:ci id="id85844">′</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> holds as stated. Moreover, if
									<m:math id="id85848" display="inline">
										<m:semantics id="id85852">
											<m:mn id="id85853">0</m:mn>
											<m:annotation-xml id="id85855" encoding="MathML-Content">
												<m:cn id="id85858">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is not a regular value of <m:math id="id85862" display="inline">
										<m:semantics id="id85865">
											<m:msub id="id85866">
												<m:mi id="id85867">μ</m:mi>
												<m:mi id="id85870">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id85872" encoding="MathML-Content">
												<m:apply id="id85875">
													<m:csymbol id="id85876" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id85881">μ</m:ci>
													<m:ci id="id85883">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, we can remove the
non-orbifold points and apply the above reasoning.</p>
							</div>

						</div>

						<div class="subsection" id="S6.SS4">
							<h3 id="id85897" class="title subsection-title">
								<span id="id85900" class="tag">6.4 </span>Replacing <m:math id="id85906" display="inline">
									<m:semantics id="id85909">
										<m:mi id="id85910">T</m:mi>
										<m:annotation-xml id="id85913" encoding="MathML-Content">
											<m:ci id="id85916">T</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by a full-rank subgroup</h3>
							<div class="para" id="S6.SS4.p1">
								<p id="id85927" style="" class="p">Suppose <m:math id="id85934" display="inline">
										<m:semantics id="id85937">
											<m:mrow id="id85938">
												<m:mi id="id85939">H</m:mi>
												<m:mo id="id85941">⊂</m:mo>
												<m:mi id="id85944">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id85946" encoding="MathML-Content">
												<m:apply id="id85949">
													<m:ci id="id85950">⊂</m:ci>
													<m:ci id="id85953">H</m:ci>
													<m:ci id="id85955">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a connected closed subgroup which contains a
maximal torus <m:math id="id85959" display="inline">
										<m:semantics id="id85962">
											<m:mi id="id85963">T</m:mi>
											<m:annotation-xml id="id85965" encoding="MathML-Content">
												<m:ci id="id85968">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Many of the results of this paper generalize in a
straightforward way to give relationships between <m:math id="id85973" display="inline">
										<m:semantics id="id85976">
											<m:mrow id="id85977">
												<m:mi id="id85978">X</m:mi>
												<m:mo id="id85980">/</m:mo>
												<m:mo id="id85982">/</m:mo>
												<m:mi id="id85984">G</m:mi>
											</m:mrow>
											<m:annotation-xml id="id85987" encoding="MathML-Content">
												<m:ci id="id85990">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id85993" display="inline">
										<m:semantics id="id85997">
											<m:mrow id="id85998">
												<m:mi id="id85999">X</m:mi>
												<m:mo id="id86001">/</m:mo>
												<m:mo id="id86003">/</m:mo>
												<m:mi id="id86005">H</m:mi>
											</m:mrow>
											<m:annotation-xml id="id86007" encoding="MathML-Content">
												<m:ci id="id86011">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S6.SS4.p2">
								<p id="id86023" style="" class="p">Denoting by <m:math id="id86029" display="inline">
										<m:semantics id="id86032">
											<m:mi id="id86034" mathvariant="fraktur">h</m:mi>
											<m:annotation-xml id="id86038" encoding="MathML-Content">
												<m:ci id="id86041">h</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> the Lie algebra of <m:math id="id86045" display="inline">
										<m:semantics id="id86048">
											<m:mi id="id86049">H</m:mi>
											<m:annotation-xml id="id86051" encoding="MathML-Content">
												<m:ci id="id86055">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, then moment map <m:math id="id86058" display="inline">
										<m:semantics id="id86061">
											<m:msub id="id86062">
												<m:mi id="id86064">μ</m:mi>
												<m:mi id="id86066">H</m:mi>
											</m:msub>
											<m:annotation-xml id="id86068" encoding="MathML-Content">
												<m:apply id="id86071">
													<m:csymbol id="id86072" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id86077">μ</m:ci>
													<m:ci id="id86080">H</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> for
the <m:math id="id86083" display="inline">
										<m:semantics id="id86086">
											<m:mi id="id86087">H</m:mi>
											<m:annotation-xml id="id86090" encoding="MathML-Content">
												<m:ci id="id86093">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action is defined analogously to <m:math id="id86096" display="inline">
										<m:semantics id="id86100">
											<m:msub id="id86101">
												<m:mi id="id86102">μ</m:mi>
												<m:mi id="id86104">T</m:mi>
											</m:msub>
											<m:annotation-xml id="id86106" encoding="MathML-Content">
												<m:apply id="id86110">
													<m:csymbol id="id86111" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id86116">μ</m:ci>
													<m:ci id="id86118">T</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, by composing <m:math id="id86121" display="inline">
										<m:semantics id="id86125">
											<m:msub id="id86126">
												<m:mi id="id86127">μ</m:mi>
												<m:mi id="id86129">G</m:mi>
											</m:msub>
											<m:annotation-xml id="id86131" encoding="MathML-Content">
												<m:apply id="id86135">
													<m:csymbol id="id86136" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id86140">μ</m:ci>
													<m:ci id="id86143">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
with the natural projection <m:math id="id86146" display="inline">
										<m:semantics id="id86150">
											<m:mrow id="id86151">
												<m:msup id="id86152">
													<m:mi id="id86153" mathvariant="fraktur">g</m:mi>
													<m:mo id="id86157">*</m:mo>
												</m:msup>
												<m:mo id="id86160">↠</m:mo>
												<m:msup id="id86162">
													<m:mi id="id86163" mathvariant="fraktur">h</m:mi>
													<m:mo id="id86167">*</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id86170" encoding="MathML-Content">
												<m:apply id="id86173">
													<m:ci id="id86174">↠</m:ci>
													<m:apply id="id86176">
														<m:csymbol id="id86177" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id86182">g</m:ci>
														<m:times id="id86184"/>
													</m:apply>
													<m:apply id="id86185">
														<m:csymbol id="id86186" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id86191">h</m:ci>
														<m:times id="id86193"/>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. We assume for simplicity
that <m:math id="id86196" display="inline">
										<m:semantics id="id86199">
											<m:msub id="id86200">
												<m:mi id="id86201">μ</m:mi>
												<m:mi id="id86204">H</m:mi>
											</m:msub>
											<m:annotation-xml id="id86206" encoding="MathML-Content">
												<m:apply id="id86209">
													<m:csymbol id="id86210" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id86215">μ</m:ci>
													<m:ci id="id86217">H</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is proper and has <m:math id="id86221" display="inline">
										<m:semantics id="id86224">
											<m:mn id="id86225">0</m:mn>
											<m:annotation-xml id="id86227" encoding="MathML-Content">
												<m:cn id="id86230">0</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math> as a regular value, and that the
									<m:math id="id86234" display="inline">
										<m:semantics id="id86237">
											<m:mi id="id86238">G</m:mi>
											<m:annotation-xml id="id86241" encoding="MathML-Content">
												<m:ci id="id86244">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action is free on <m:math id="id86247" display="inline">
										<m:semantics id="id86251">
											<m:mrow id="id86252">
												<m:msubsup id="id86253">
													<m:mi id="id86254">μ</m:mi>
													<m:mi id="id86256">G</m:mi>
													<m:mrow id="id86258">
														<m:mo id="id86260">-</m:mo>
														<m:mn id="id86262">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id86264">⁢</m:mo>
												<m:mfenced id="id86266" open="(" close=")">
													<m:mn id="id86271">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id86273" encoding="MathML-Content">
												<m:apply id="id86277">
													<m:times id="id86278"/>
													<m:apply id="id86279">
														<m:csymbol id="id86280" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id86285">
															<m:csymbol id="id86286" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id86290">μ</m:ci>
															<m:ci id="id86293">G</m:ci>
														</m:apply>
														<m:apply id="id86295">
															<m:minus id="id86296"/>
															<m:cn id="id86297">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id86299">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the <m:math id="id86303" display="inline">
										<m:semantics id="id86306">
											<m:mi id="id86307">H</m:mi>
											<m:annotation-xml id="id86309" encoding="MathML-Content">
												<m:ci id="id86312">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action is free on
									<m:math id="id86316" display="inline">
										<m:semantics id="id86319">
											<m:mrow id="id86320">
												<m:msubsup id="id86321">
													<m:mi id="id86322">μ</m:mi>
													<m:mi id="id86325">H</m:mi>
													<m:mrow id="id86327">
														<m:mo id="id86328">-</m:mo>
														<m:mn id="id86330">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id86332">⁢</m:mo>
												<m:mfenced id="id86335" open="(" close=")">
													<m:mn id="id86340">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id86342" encoding="MathML-Content">
												<m:apply id="id86345">
													<m:times id="id86346"/>
													<m:apply id="id86347">
														<m:csymbol id="id86348" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id86353">
															<m:csymbol id="id86354" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id86359">μ</m:ci>
															<m:ci id="id86361">H</m:ci>
														</m:apply>
														<m:apply id="id86363">
															<m:minus id="id86364"/>
															<m:cn id="id86366">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id86368">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> (there are obvious generalizations when these
conditions are not met, as described above).</p>
							</div>

							<div class="para" id="S6.SS4.p3">
								<p id="id86381" style="" class="p">Under the adjoint action of <m:math id="id86387" display="inline">
										<m:semantics id="id86390">
											<m:mi id="id86392">H</m:mi>
											<m:annotation-xml id="id86394" encoding="MathML-Content">
												<m:ci id="id86397">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, the Lie algebra <m:math id="id86400" display="inline">
										<m:semantics id="id86404">
											<m:mi id="id86405" mathvariant="fraktur">g</m:mi>
											<m:annotation-xml id="id86409" encoding="MathML-Content">
												<m:ci id="id86413">g</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> decomposes into
subrepresentations</p>
								<table class="equation" id="S6.Ex4">

									<tr valign="baseline" class="equation" id="S6.Ex4">
										<td id="id86433" class="eqpad"/>
										<td id="id86437" nowrap="yes" align="center" colspan="1">
											<m:math id="id86445" display="block">
												<m:semantics id="id86448">
													<m:mrow id="id86449">
														<m:mrow id="id86450">
															<m:mi id="id86451" mathvariant="fraktur">g</m:mi>
															<m:mo id="id86456">≅</m:mo>
															<m:mrow id="id86458">
																<m:mi id="id86459" mathvariant="fraktur">h</m:mi>
																<m:mo id="id86463">⊕</m:mo>
																<m:mi id="id86466" mathvariant="fraktur">e</m:mi>
															</m:mrow>
														</m:mrow>
														<m:mo id="id86470">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id86472" encoding="MathML-Content">
														<m:apply id="id86476">
															<m:approx id="id86477"/>
															<m:ci id="id86478">g</m:ci>
															<m:apply id="id86480">
																<m:ci id="id86481">⊕</m:ci>
																<m:ci id="id86483">h</m:ci>
																<m:ci id="id86486">e</m:ci>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id86489" class="eqpad"/>
									</tr>

								</table>
								<p id="id86495" style="" class="p">This decomposition is compatible with the <m:math id="id86502" display="inline">
										<m:semantics id="id86505">
											<m:mi id="id86506">T</m:mi>
											<m:annotation-xml id="id86508" encoding="MathML-Content">
												<m:ci id="id86512">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-action, and
									<m:math id="id86515" display="inline">
										<m:semantics id="id86519">
											<m:mrow id="id86520">
												<m:mi id="id86521" mathvariant="fraktur">t</m:mi>
												<m:mo id="id86525">⊂</m:mo>
												<m:mi id="id86528" mathvariant="fraktur">h</m:mi>
											</m:mrow>
											<m:annotation-xml id="id86532" encoding="MathML-Content">
												<m:apply id="id86535">
													<m:ci id="id86536">⊂</m:ci>
													<m:ci id="id86539">t</m:ci>
													<m:ci id="id86541">h</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, hence a choice of positive roots for <m:math id="id86544" display="inline">
										<m:semantics id="id86548">
											<m:mi id="id86549">T</m:mi>
											<m:annotation-xml id="id86551" encoding="MathML-Content">
												<m:ci id="id86554">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> gives a
complex structure to the <m:math id="id86558" display="inline">
										<m:semantics id="id86561">
											<m:mi id="id86562">H</m:mi>
											<m:annotation-xml id="id86564" encoding="MathML-Content">
												<m:ci id="id86568">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-representation <m:math id="id86571" display="inline">
										<m:semantics id="id86575">
											<m:mi id="id86576" mathvariant="fraktur">e</m:mi>
											<m:annotation-xml id="id86580" encoding="MathML-Content">
												<m:ci id="id86583">e</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.  We denote by
									<m:math id="id86587" display="inline">
										<m:semantics id="id86590">
											<m:mrow id="id86591">
												<m:mi id="id86592" mathvariant="script">E</m:mi>
												<m:mo id="id86597">→</m:mo>
												<m:mi id="id86599">X</m:mi>
												<m:mo id="id86601">/</m:mo>
												<m:mo id="id86603">/</m:mo>
												<m:mi id="id86606">H</m:mi>
											</m:mrow>
											<m:annotation-xml id="id86608" encoding="MathML-Content">
												<m:ci id="id86611">E</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> the associated vector bundle <m:math id="id86615" display="inline">
										<m:semantics id="id86618">
											<m:mrow id="id86619">
												<m:msubsup id="id86620">
													<m:mi id="id86621">μ</m:mi>
													<m:mi id="id86624">H</m:mi>
													<m:mrow id="id86626">
														<m:mo id="id86627">-</m:mo>
														<m:mn id="id86629">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id86631">(</m:mo>
												<m:mn id="id86633">0</m:mn>
												<m:mo id="id86635">)</m:mo>
												<m:msub id="id86637">
													<m:mo id="id86638">×</m:mo>
													<m:mi id="id86641">H</m:mi>
												</m:msub>
												<m:mi id="id86643" mathvariant="fraktur">e</m:mi>
												<m:mo id="id86647">→</m:mo>
												<m:mi id="id86650">X</m:mi>
												<m:mo id="id86652">/</m:mo>
												<m:mo id="id86654">/</m:mo>
												<m:mi id="id86656">H</m:mi>
											</m:mrow>
											<m:annotation-xml id="id86658" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S6.SS4.p4">
								<p id="id86672" style="" class="p">With these definitions, the main topological result generalizes in the
obvious way, with the bundle <m:math id="id86679" display="inline">
										<m:semantics id="id86682">
											<m:mrow id="id86683">
												<m:msub id="id86684">
													<m:mo id="id86685">⊕</m:mo>
													<m:mrow id="id86688">
														<m:mi id="id86689">α</m:mi>
														<m:mo id="id86691">∈</m:mo>
														<m:msup id="id86693">
															<m:mi id="id86694" mathvariant="normal">Δ</m:mi>
															<m:mo id="id86699">+</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:msub id="id86701">
													<m:mi id="id86702">L</m:mi>
													<m:mi id="id86704">α</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id86707" encoding="MathML-Content">
												<m:apply id="id86710">
													<m:apply id="id86711">
														<m:csymbol id="id86712" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id86717">⊕</m:ci>
														<m:apply id="id86719">
															<m:ci id="id86720">∈</m:ci>
															<m:ci id="id86723">α</m:ci>
															<m:apply id="id86725">
																<m:csymbol id="id86726" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id86731">Δ</m:ci>
																<m:plus id="id86733"/>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply id="id86734">
														<m:csymbol id="id86736" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id86740">L</m:ci>
														<m:ci id="id86742">α</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> replaced by <m:math id="id86746" display="inline">
										<m:semantics id="id86749">
											<m:mi id="id86750" mathvariant="script">E</m:mi>
											<m:annotation-xml id="id86755" encoding="MathML-Content">
												<m:ci id="id86758">E</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and
									<m:math id="id86762" display="inline">
										<m:semantics id="id86765">
											<m:mrow id="id86766">
												<m:msub id="id86767">
													<m:mo id="id86768">⊕</m:mo>
													<m:mrow id="id86771">
														<m:mi id="id86772">α</m:mi>
														<m:mo id="id86774">∈</m:mo>
														<m:msup id="id86776">
															<m:mi id="id86778" mathvariant="normal">Δ</m:mi>
															<m:mo id="id86782">-</m:mo>
														</m:msup>
													</m:mrow>
												</m:msub>
												<m:msub id="id86784">
													<m:mi id="id86785">L</m:mi>
													<m:mi id="id86788">α</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id86790" encoding="MathML-Content">
												<m:apply id="id86793">
													<m:apply id="id86794">
														<m:csymbol id="id86795" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id86800">⊕</m:ci>
														<m:apply id="id86802">
															<m:ci id="id86804">∈</m:ci>
															<m:ci id="id86806">α</m:ci>
															<m:apply id="id86808">
																<m:csymbol id="id86809" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id86814">Δ</m:ci>
																<m:minus id="id86816"/>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:apply id="id86818">
														<m:csymbol id="id86819" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id86823">L</m:ci>
														<m:ci id="id86825">α</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> replaced by the dual bundle <m:math id="id86829" display="inline">
										<m:semantics id="id86833">
											<m:msup id="id86834">
												<m:mi id="id86835" mathvariant="script">E</m:mi>
												<m:mo id="id86839">*</m:mo>
											</m:msup>
											<m:annotation-xml id="id86841" encoding="MathML-Content">
												<m:apply id="id86845">
													<m:csymbol id="id86846" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id86850">E</m:ci>
													<m:times id="id86852"/>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Thus, with the obvious
definition of a lift of a cohomology class or vector bundle, we have</p>
							</div>

							<div class="theorem theoremBH" id="ThmtheoremBHx1">
								<h6 id="id86864" class="title theorem-title bold">
									<span id="id86868" class="tag">Theorem B<m:math id="id86872" display="inline">
											<m:semantics id="id86876">
												<m:msub id="id86877">
													<m:mi id="id86878"/>
													<m:mi id="id86879">H</m:mi>
												</m:msub>
												<m:annotation-xml id="id86881" encoding="MathML-Content">
													<m:apply id="id86884">
														<m:ci id="id86885">H</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</span> (Integration formula).</h6>
								<div class="para" id="ThmtheoremBHx1.p1">
									<p id="id86896" style="" class="p">
										<span id="id86902" style="" class="text italic">Given a cohomology class  <m:math id="id86908" display="inline">
												<m:semantics id="id86912">
													<m:mrow id="id86913">
														<m:mi id="id86914">a</m:mi>
														<m:mo id="id86916" mathvariant="normal">∈</m:mo>
														<m:msup id="id86921">
															<m:mi id="id86922">H</m:mi>
															<m:mo id="id86924" mathvariant="normal">*</m:mo>
														</m:msup>
														<m:mo id="id86928" mathvariant="normal">(</m:mo>
														<m:mi id="id86933">X</m:mi>
														<m:mo id="id86935" mathvariant="normal">/</m:mo>
														<m:mo id="id86939" mathvariant="normal">/</m:mo>
														<m:mi id="id86944">G</m:mi>
														<m:mo id="id86946" mathvariant="normal">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id86950" encoding="MathML-Content"/>
												</m:semantics>
											</m:math> with lift <m:math id="id86955" display="inline">
												<m:semantics id="id86958">
													<m:mrow id="id86959">
														<m:mover id="id86960" accent="true">
															<m:mi id="id86964">a</m:mi>
															<m:mo id="id86966" mathvariant="normal">~</m:mo>
														</m:mover>
														<m:mo id="id86970" mathvariant="normal">∈</m:mo>
														<m:msup id="id86975">
															<m:mi id="id86976">H</m:mi>
															<m:mo id="id86978" mathvariant="normal">*</m:mo>
														</m:msup>
														<m:mo id="id86982" mathvariant="normal">(</m:mo>
														<m:mi id="id86987">X</m:mi>
														<m:mo id="id86989" mathvariant="normal">/</m:mo>
														<m:mo id="id86993" mathvariant="normal">/</m:mo>
														<m:mi id="id86998">H</m:mi>
														<m:mo id="id87000" mathvariant="normal">)</m:mo>
													</m:mrow>
													<m:annotation-xml id="id87004" encoding="MathML-Content"/>
												</m:semantics>
											</m:math>, then</span>
									</p>
									<table class="equation" id="S6.Ex5">

										<tr valign="baseline" class="equation" id="S6.Ex5">
											<td id="id87026" class="eqpad"/>
											<td id="id87030" nowrap="yes" align="center" colspan="1">
												<m:math id="id87037" display="block">
													<m:semantics id="id87041">
														<m:mrow id="id87042">
															<m:mrow id="id87043">
																<m:mrow id="id87044">
																	<m:msub id="id87045">
																		<m:mo id="id87046">∫</m:mo>
																		<m:mrow id="id87048">
																			<m:mi id="id87049">X</m:mi>
																			<m:mo id="id87052">⁣</m:mo>
																			<m:mo id="id87054">/</m:mo>
																			<m:mo id="id87056">⁣</m:mo>
																			<m:mrow id="id87058">
																				<m:none id="id87060"/>
																				<m:mo id="id87061">/</m:mo>
																				<m:mi id="id87063">G</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msub>
																	<m:mi id="id87065">a</m:mi>
																</m:mrow>
																<m:mo id="id87067">=</m:mo>
																<m:mrow id="id87069">
																	<m:mfrac id="id87070">
																		<m:mfenced id="id87071" open="|" close="|">
																			<m:mrow id="id87076">
																				<m:mi id="id87077">W</m:mi>
																				<m:mo id="id87080">⁢</m:mo>
																				<m:mfenced id="id87082" open="(" close=")">
																					<m:mi id="id87087">H</m:mi>
																				</m:mfenced>
																			</m:mrow>
																		</m:mfenced>
																		<m:mfenced id="id87089" open="|" close="|">
																			<m:mrow id="id87094">
																				<m:mi id="id87095">W</m:mi>
																				<m:mo id="id87097">⁢</m:mo>
																				<m:mfenced id="id87100" open="(" close=")">
																					<m:mi id="id87105">G</m:mi>
																				</m:mfenced>
																			</m:mrow>
																		</m:mfenced>
																	</m:mfrac>
																	<m:mo id="id87107">⁢</m:mo>
																	<m:mrow id="id87109">
																		<m:msub id="id87110">
																			<m:mo id="id87112">∫</m:mo>
																			<m:mrow id="id87114">
																				<m:mi id="id87115">X</m:mi>
																				<m:mo id="id87117">⁣</m:mo>
																				<m:mo id="id87120">/</m:mo>
																				<m:mo id="id87122">⁣</m:mo>
																				<m:mrow id="id87124">
																					<m:none id="id87125"/>
																					<m:mo id="id87126">/</m:mo>
																					<m:mi id="id87128">H</m:mi>
																				</m:mrow>
																			</m:mrow>
																		</m:msub>
																		<m:mover id="id87130" accent="true">
																			<m:mi id="id87134">a</m:mi>
																			<m:mo id="id87136">~</m:mo>
																		</m:mover>
																	</m:mrow>
																</m:mrow>
																<m:mo id="id87138">⌣</m:mo>
																<m:mrow id="id87140">
																	<m:mi id="id87142">e</m:mi>
																	<m:mo id="id87144">⁢</m:mo>
																	<m:mfenced id="id87146" open="(" close=")">
																		<m:mrow id="id87151">
																			<m:mi id="id87152" mathvariant="script">E</m:mi>
																			<m:mo id="id87157">⊕</m:mo>
																			<m:msup id="id87159">
																				<m:mi id="id87160" mathvariant="script">E</m:mi>
																				<m:mo id="id87164">*</m:mo>
																			</m:msup>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:mrow>
															<m:mo id="id87167">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id87169" encoding="MathML-Content">
															<m:apply id="id87172">
																<m:ci id="id87173"/>
																<m:apply id="id87174">
																	<m:apply id="id87175">
																		<m:csymbol id="id87176" cd="ambiguous">subscript</m:csymbol>
																		<m:int id="id87181"/>
																		<m:apply id="id87182">
																			<m:list id="id87183"/>
																			<m:ci id="id87184">X</m:ci>
																			<m:divide id="id87186"/>
																			<m:apply id="id87187">
																				<m:divide id="id87188"/>
																				<m:ci id="id87190"/>
																				<m:ci id="id87191">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:ci id="id87193">a</m:ci>
																</m:apply>
																<m:eq id="id87195"/>
																<m:apply id="id87196">
																	<m:times id="id87197"/>
																	<m:apply id="id87198">
																		<m:divide id="id87199"/>
																		<m:apply id="id87200">
																			<m:abs id="id87201"/>
																			<m:apply id="id87202">
																				<m:times id="id87203"/>
																				<m:ci id="id87204">W</m:ci>
																				<m:ci id="id87207">H</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id87209">
																			<m:abs id="id87210"/>
																			<m:apply id="id87211">
																				<m:times id="id87212"/>
																				<m:ci id="id87213">W</m:ci>
																				<m:ci id="id87215">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id87217">
																		<m:apply id="id87218">
																			<m:csymbol id="id87219" cd="ambiguous">subscript</m:csymbol>
																			<m:int id="id87224"/>
																			<m:apply id="id87225">
																				<m:list id="id87226"/>
																				<m:ci id="id87227">X</m:ci>
																				<m:divide id="id87229"/>
																				<m:apply id="id87230">
																					<m:divide id="id87232"/>
																					<m:ci id="id87233"/>
																					<m:ci id="id87234">H</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id87236">
																			<m:ci id="id87237">~</m:ci>
																			<m:ci id="id87239">a</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:ci id="id87241">⌣</m:ci>
																<m:apply id="id87244">
																	<m:times id="id87245"/>
																	<m:ci id="id87246">e</m:ci>
																	<m:apply id="id87248">
																		<m:ci id="id87249">⊕</m:ci>
																		<m:ci id="id87251">E</m:ci>
																		<m:apply id="id87253">
																			<m:csymbol id="id87254" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id87259">E</m:ci>
																			<m:times id="id87261"/>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id87263" class="eqpad"/>
										</tr>

									</table>
									<p id="id87270" style="" class="p">
										<span id="id87275" style="" class="text italic">where <m:math id="id87282" display="inline">
												<m:semantics id="id87285">
													<m:mrow id="id87286">
														<m:mi id="id87287">W</m:mi>
														<m:mo id="id87289" mathvariant="italic">⁢</m:mo>
														<m:mfenced id="id87294" open="(" close=")">
															<m:mi id="id87299">H</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:annotation-xml id="id87301" encoding="MathML-Content">
														<m:apply id="id87304">
															<m:times id="id87306"/>
															<m:ci id="id87307">W</m:ci>
															<m:ci id="id87309">H</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id87312" display="inline">
												<m:semantics id="id87316">
													<m:mrow id="id87317">
														<m:mi id="id87318">W</m:mi>
														<m:mo id="id87320" mathvariant="italic">⁢</m:mo>
														<m:mfenced id="id87324" open="(" close=")">
															<m:mi id="id87330">G</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:annotation-xml id="id87332" encoding="MathML-Content">
														<m:apply id="id87335">
															<m:times id="id87336"/>
															<m:ci id="id87337">W</m:ci>
															<m:ci id="id87339">G</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> are the Weyl groups of <m:math id="id87343" display="inline">
												<m:semantics id="id87346">
													<m:mi id="id87347">H</m:mi>
													<m:annotation-xml id="id87349" encoding="MathML-Content">
														<m:ci id="id87353">H</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> and <m:math id="id87356" display="inline">
												<m:semantics id="id87360">
													<m:mi id="id87361">G</m:mi>
													<m:annotation-xml id="id87363" encoding="MathML-Content">
														<m:ci id="id87366">G</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> respectively.</span>
									</p>
								</div>

							</div>

							<div class="theorem theoremCH" id="ThmtheoremCHx1">
								<h6 id="id87379" class="title theorem-title bold">
									<span id="id87383" class="tag">Theorem C<m:math id="id87387" display="inline">
											<m:semantics id="id87391">
												<m:msub id="id87392">
													<m:mi id="id87393"/>
													<m:mi id="id87394">H</m:mi>
												</m:msub>
												<m:annotation-xml id="id87396" encoding="MathML-Content">
													<m:apply id="id87399">
														<m:ci id="id87400">H</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</span> (Index formula).</h6>
								<div class="para" id="ThmtheoremCHx1.p1">
									<p id="id87412" style="" class="p">
										<span id="id87417" style="" class="text italic">Suppose <m:math id="id87423" display="inline">
												<m:semantics id="id87427">
													<m:mrow id="id87428">
														<m:mi id="id87429">V</m:mi>
														<m:mo id="id87431" mathvariant="normal">→</m:mo>
														<m:mi id="id87436">X</m:mi>
														<m:mo id="id87438" mathvariant="normal">/</m:mo>
														<m:mo id="id87442" mathvariant="normal">/</m:mo>
														<m:mi id="id87446">G</m:mi>
													</m:mrow>
													<m:annotation-xml id="id87449" encoding="MathML-Content">
														<m:ci id="id87452">V</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math> is a complex vector bundle, and <m:math id="id87456" display="inline">
												<m:semantics id="id87459">
													<m:mrow id="id87460">
														<m:mover id="id87461" accent="true">
															<m:mi id="id87464">V</m:mi>
															<m:mo id="id87466" mathvariant="normal">~</m:mo>
														</m:mover>
														<m:mo id="id87471" mathvariant="normal">→</m:mo>
														<m:mi id="id87476">X</m:mi>
														<m:mo id="id87478" mathvariant="normal">/</m:mo>
														<m:mo id="id87482" mathvariant="normal">/</m:mo>
														<m:mi id="id87486">H</m:mi>
													</m:mrow>
													<m:annotation-xml id="id87489" encoding="MathML-Content">
														<m:apply id="id87492">
															<m:ci id="id87493">~</m:ci>
															<m:ci id="id87495">V</m:ci>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> is a lift of <m:math id="id87499" display="inline">
												<m:semantics id="id87502">
													<m:mi id="id87503">V</m:mi>
													<m:annotation-xml id="id87505" encoding="MathML-Content">
														<m:ci id="id87508">V</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>. Then</span>
									</p>
									<table class="equation" id="S6.Ex6">

										<tr valign="baseline" class="equation" id="S6.Ex6">
											<td id="id87529" class="eqpad"/>
											<td id="id87533" nowrap="yes" align="center" colspan="1">
												<m:math id="id87540" display="block">
													<m:semantics id="id87544">
														<m:mrow id="id87545">
															<m:mrow id="id87546">
																<m:msup id="id87547">
																	<m:mrow id="id87548">
																		<m:mi id="id87549">i</m:mi>
																		<m:mo id="id87551">⁢</m:mo>
																		<m:mi id="id87554">n</m:mi>
																		<m:mo id="id87556">⁢</m:mo>
																		<m:mi id="id87558">d</m:mi>
																		<m:mo id="id87560">⁢</m:mo>
																		<m:mi id="id87563">e</m:mi>
																		<m:mo id="id87565">⁢</m:mo>
																		<m:mi id="id87567">x</m:mi>
																	</m:mrow>
																	<m:mrow id="id87569">
																		<m:mi id="id87570">X</m:mi>
																		<m:mo id="id87572">⁣</m:mo>
																		<m:mo id="id87575">/</m:mo>
																		<m:mo id="id87577">⁣</m:mo>
																		<m:mrow id="id87579">
																			<m:none id="id87580"/>
																			<m:mo id="id87582">/</m:mo>
																			<m:mi id="id87584">G</m:mi>
																		</m:mrow>
																	</m:mrow>
																</m:msup>
																<m:mo id="id87586">⁢</m:mo>
																<m:msub id="id87588">
																	<m:mi id="id87589">D</m:mi>
																	<m:mi id="id87591">V</m:mi>
																</m:msub>
															</m:mrow>
															<m:mo id="id87594">=</m:mo>
															<m:mrow id="id87596">
																<m:mrow id="id87597">
																	<m:msup id="id87598">
																		<m:mrow id="id87599">
																			<m:mi id="id87600">i</m:mi>
																			<m:mo id="id87602">⁢</m:mo>
																			<m:mi id="id87604">n</m:mi>
																			<m:mo id="id87607">⁢</m:mo>
																			<m:mi id="id87609">d</m:mi>
																			<m:mo id="id87611">⁢</m:mo>
																			<m:mi id="id87614">e</m:mi>
																			<m:mo id="id87616">⁢</m:mo>
																			<m:mi id="id87618">x</m:mi>
																		</m:mrow>
																		<m:mrow id="id87620">
																			<m:mi id="id87621">X</m:mi>
																			<m:mo id="id87623">⁣</m:mo>
																			<m:mo id="id87626">/</m:mo>
																			<m:mo id="id87628">⁣</m:mo>
																			<m:mrow id="id87630">
																				<m:none id="id87631"/>
																				<m:mo id="id87632">/</m:mo>
																				<m:mi id="id87635">H</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msup>
																	<m:mo id="id87637">⁢</m:mo>
																	<m:msub id="id87639">
																		<m:mi id="id87640">D</m:mi>
																		<m:mrow id="id87642">
																			<m:mrow id="id87643">
																				<m:mover id="id87644" accent="true">
																					<m:mi id="id87648">V</m:mi>
																					<m:mo id="id87650">~</m:mo>
																				</m:mover>
																				<m:mo id="id87652">⊗</m:mo>
																				<m:msup id="id87654">
																					<m:mi id="id87656" mathvariant="normal">Λ</m:mi>
																					<m:mtext id="id87660" mathvariant="italic">even</m:mtext>
																				</m:msup>
																			</m:mrow>
																			<m:mo id="id87665">⁢</m:mo>
																			<m:mi id="id87667" mathvariant="script">E</m:mi>
																		</m:mrow>
																	</m:msub>
																</m:mrow>
																<m:mo id="id87672">-</m:mo>
																<m:mrow id="id87674">
																	<m:msup id="id87675">
																		<m:mrow id="id87676">
																			<m:mi id="id87677">i</m:mi>
																			<m:mo id="id87679">⁢</m:mo>
																			<m:mi id="id87682">n</m:mi>
																			<m:mo id="id87684">⁢</m:mo>
																			<m:mi id="id87686">d</m:mi>
																			<m:mo id="id87688">⁢</m:mo>
																			<m:mi id="id87691">e</m:mi>
																			<m:mo id="id87693">⁢</m:mo>
																			<m:mi id="id87695">x</m:mi>
																		</m:mrow>
																		<m:mrow id="id87697">
																			<m:mi id="id87698">X</m:mi>
																			<m:mo id="id87700">⁣</m:mo>
																			<m:mo id="id87703">/</m:mo>
																			<m:mo id="id87705">⁣</m:mo>
																			<m:mrow id="id87707">
																				<m:none id="id87708"/>
																				<m:mo id="id87710">/</m:mo>
																				<m:mi id="id87712">T</m:mi>
																			</m:mrow>
																		</m:mrow>
																	</m:msup>
																	<m:mo id="id87714">⁢</m:mo>
																	<m:msub id="id87716">
																		<m:mi id="id87717">D</m:mi>
																		<m:mrow id="id87719">
																			<m:mrow id="id87720">
																				<m:mover id="id87722" accent="true">
																					<m:mi id="id87725">V</m:mi>
																					<m:mo id="id87727">~</m:mo>
																				</m:mover>
																				<m:mo id="id87729">⊗</m:mo>
																				<m:msup id="id87732">
																					<m:mi id="id87733" mathvariant="normal">Λ</m:mi>
																					<m:mtext id="id87737" mathvariant="italic">odd</m:mtext>
																				</m:msup>
																			</m:mrow>
																			<m:mo id="id87742">⁢</m:mo>
																			<m:mi id="id87744" mathvariant="script">E</m:mi>
																		</m:mrow>
																	</m:msub>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:annotation-xml id="id87748" encoding="MathML-Content">
															<m:apply id="id87752">
																<m:eq id="id87753"/>
																<m:apply id="id87754">
																	<m:times id="id87755"/>
																	<m:apply id="id87756">
																		<m:csymbol id="id87757" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id87762"/>
																		<m:apply id="id87763">
																			<m:list id="id87764"/>
																			<m:ci id="id87765">X</m:ci>
																			<m:divide id="id87767"/>
																			<m:apply id="id87768">
																				<m:divide id="id87769"/>
																				<m:ci id="id87770"/>
																				<m:ci id="id87771">G</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id87774">
																		<m:csymbol id="id87775" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id87779">D</m:ci>
																		<m:ci id="id87781">V</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id87784">
																	<m:minus id="id87785"/>
																	<m:apply id="id87786">
																		<m:times id="id87787"/>
																		<m:apply id="id87788">
																			<m:csymbol id="id87789" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id87794"/>
																			<m:apply id="id87795">
																				<m:list id="id87796"/>
																				<m:ci id="id87797">X</m:ci>
																				<m:divide id="id87799"/>
																				<m:apply id="id87800">
																					<m:divide id="id87801"/>
																					<m:ci id="id87802"/>
																					<m:ci id="id87803">H</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id87805">
																			<m:csymbol id="id87806" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id87811">D</m:ci>
																			<m:apply id="id87813">
																				<m:times id="id87814"/>
																				<m:apply id="id87815">
																					<m:ci id="id87816">⊗</m:ci>
																					<m:apply id="id87819">
																						<m:ci id="id87820">~</m:ci>
																						<m:ci id="id87822">V</m:ci>
																					</m:apply>
																					<m:apply id="id87824">
																						<m:csymbol id="id87825" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id87830">Λ</m:ci>
																						<m:mtext id="id87832">even</m:mtext>
																					</m:apply>
																				</m:apply>
																				<m:ci id="id87835">E</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id87837">
																		<m:times id="id87838"/>
																		<m:apply id="id87839">
																			<m:csymbol id="id87840" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id87845"/>
																			<m:apply id="id87846">
																				<m:list id="id87847"/>
																				<m:ci id="id87848">X</m:ci>
																				<m:divide id="id87850"/>
																				<m:apply id="id87851">
																					<m:divide id="id87852"/>
																					<m:ci id="id87853"/>
																					<m:ci id="id87854">T</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id87856">
																			<m:csymbol id="id87857" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id87862">D</m:ci>
																			<m:apply id="id87864">
																				<m:times id="id87865"/>
																				<m:apply id="id87866">
																					<m:ci id="id87867">⊗</m:ci>
																					<m:apply id="id87870">
																						<m:ci id="id87871">~</m:ci>
																						<m:ci id="id87873">V</m:ci>
																					</m:apply>
																					<m:apply id="id87875">
																						<m:csymbol id="id87876" cd="ambiguous">superscript</m:csymbol>
																						<m:ci id="id87881">Λ</m:ci>
																						<m:mtext id="id87883">odd</m:mtext>
																					</m:apply>
																				</m:apply>
																				<m:ci id="id87885">E</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:annotation-xml>
													</m:semantics>
												</m:math>
											</td>
											<td id="id87889" class="eqpad"/>
										</tr>

									</table>
								</div>

							</div>

							<div class="para" id="S6.SS4.p5">
								<p id="id87903" style="" class="p">Theorem A generalizes in a special case. Suppose <m:math id="id87910" display="inline">
										<m:semantics id="id87913">
											<m:mrow id="id87914">
												<m:mi id="id87915">W</m:mi>
												<m:mo id="id87918">⁢</m:mo>
												<m:mfenced id="id87920" open="(" close=")">
													<m:mi id="id87925">H</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id87927" encoding="MathML-Content">
												<m:apply id="id87930">
													<m:times id="id87932"/>
													<m:ci id="id87933">W</m:ci>
													<m:ci id="id87935">H</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a
									<span id="id87938" style="" class="text italic">normal</span> subgroup of <m:math id="id87946" display="inline">
										<m:semantics id="id87950">
											<m:mrow id="id87951">
												<m:mi id="id87952">W</m:mi>
												<m:mo id="id87954">⁢</m:mo>
												<m:mfenced id="id87956" open="(" close=")">
													<m:mi id="id87961">G</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id87963" encoding="MathML-Content">
												<m:apply id="id87967">
													<m:times id="id87968"/>
													<m:ci id="id87969">W</m:ci>
													<m:ci id="id87971">G</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Then the quotient group
									<m:math id="id87975" display="inline">
										<m:semantics id="id87978">
											<m:mrow id="id87979">
												<m:mrow id="id87980">
													<m:mrow id="id87981">
														<m:mi id="id87982">W</m:mi>
														<m:mo id="id87984">⁢</m:mo>
														<m:mfenced id="id87987" open="(" close=")">
															<m:mi id="id87992">G</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id87994">/</m:mo>
													<m:mi id="id87996">W</m:mi>
												</m:mrow>
												<m:mo id="id87998">⁢</m:mo>
												<m:mfenced id="id88001" open="(" close=")">
													<m:mi id="id88006">H</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id88008" encoding="MathML-Content">
												<m:apply id="id88011">
													<m:times id="id88012"/>
													<m:apply id="id88013">
														<m:divide id="id88014"/>
														<m:apply id="id88015">
															<m:times id="id88016"/>
															<m:ci id="id88018">W</m:ci>
															<m:ci id="id88020">G</m:ci>
														</m:apply>
														<m:ci id="id88022">W</m:ci>
													</m:apply>
													<m:ci id="id88024">H</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> can be thought of as the relative Weyl group for <m:math id="id88028" display="inline">
										<m:semantics id="id88031">
											<m:mi id="id88032">H</m:mi>
											<m:annotation-xml id="id88034" encoding="MathML-Content">
												<m:ci id="id88038">H</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in
									<m:math id="id88041" display="inline">
										<m:semantics id="id88044">
											<m:mi id="id88045">G</m:mi>
											<m:annotation-xml id="id88048" encoding="MathML-Content">
												<m:ci id="id88051">G</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and <m:math id="id88054" display="inline">
										<m:semantics id="id88058">
											<m:mrow id="id88059">
												<m:mi id="id88060">X</m:mi>
												<m:mo id="id88062">/</m:mo>
												<m:mo id="id88064">/</m:mo>
												<m:mi id="id88066">H</m:mi>
											</m:mrow>
											<m:annotation-xml id="id88068" encoding="MathML-Content">
												<m:ci id="id88072">X</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> carries an action of this relative Weyl group.
In this case we have</p>
							</div>

							<div class="theorem theoremAH" id="ThmtheoremAHx1">
								<h6 id="id88084" class="title theorem-title bold">
									<span id="id88088" class="tag">Theorem A<m:math id="id88092" display="inline">
											<m:semantics id="id88096">
												<m:msub id="id88097">
													<m:mi id="id88098"/>
													<m:mi id="id88099">H</m:mi>
												</m:msub>
												<m:annotation-xml id="id88101" encoding="MathML-Content">
													<m:apply id="id88104">
														<m:ci id="id88105">H</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>
									</span> (Cohomology rings).</h6>
								<div class="para" id="ThmtheoremAHx1.p1">
									<p id="id88116" style="" class="p">
										<span id="id88122" style="" class="text italic">There is a natural ring isomorphism</span>
									</p>
									<table class="equation" id="S6.Ex7">

										<tr valign="baseline" class="equation" id="S6.Ex7">
											<td id="id88145" class="eqpad"/>
											<td id="id88149" nowrap="yes" align="center" colspan="1">
												<m:math id="id88157" display="block">
													<m:semantics id="id88160">
														<m:mrow id="id88161">
															<m:msup id="id88162">
																<m:mi id="id88163">H</m:mi>
																<m:mo id="id88165">*</m:mo>
															</m:msup>
															<m:mo id="id88168">(</m:mo>
															<m:mi id="id88170">X</m:mi>
															<m:mo id="id88172">/</m:mo>
															<m:mo id="id88174">/</m:mo>
															<m:mi id="id88176">G</m:mi>
															<m:mo id="id88178">;</m:mo>
															<m:mi id="id88180" mathvariant="double-struck">Q</m:mi>
															<m:mo id="id88185">)</m:mo>
															<m:mo id="id88187">≅</m:mo>
															<m:mfrac id="id88189">
																<m:mrow id="id88190">
																	<m:msup id="id88191">
																		<m:mi id="id88192">H</m:mi>
																		<m:mo id="id88195">*</m:mo>
																	</m:msup>
																	<m:mo id="id88197">(</m:mo>
																	<m:mi id="id88199">X</m:mi>
																	<m:mo id="id88201">/</m:mo>
																	<m:mo id="id88203">/</m:mo>
																	<m:mi id="id88205">H</m:mi>
																	<m:mo id="id88207">;</m:mo>
																	<m:mi id="id88210" mathvariant="double-struck">Q</m:mi>
																	<m:msup id="id88214">
																		<m:mo id="id88215">)</m:mo>
																		<m:mi id="id88217">W</m:mi>
																	</m:msup>
																</m:mrow>
																<m:mrow id="id88219">
																	<m:mo id="id88220">ann</m:mo>
																	<m:mo id="id88222">⁡</m:mo>
																	<m:mfenced id="id88225" open="(" close=")">
																		<m:mrow id="id88230">
																			<m:mi id="id88231">e</m:mi>
																			<m:mo id="id88233">⁢</m:mo>
																			<m:mfenced id="id88236" open="(" close=")">
																				<m:mrow id="id88241">
																					<m:mi id="id88242" mathvariant="script">E</m:mi>
																					<m:mo id="id88246">⊕</m:mo>
																					<m:msup id="id88248">
																						<m:mi id="id88250" mathvariant="script">E</m:mi>
																						<m:mo id="id88254">*</m:mo>
																					</m:msup>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:mfrac>
															<m:mo id="id88256">,</m:mo>
														</m:mrow>
														<m:annotation-xml id="id88258" encoding="MathML-Content"/>
													</m:semantics>
												</m:math>
											</td>
											<td id="id88263" class="eqpad"/>
										</tr>

									</table>
									<p id="id88269" style="" class="p">
										<span id="id88274" style="" class="text italic">where <m:math id="id88281" display="inline">
												<m:semantics id="id88284">
													<m:mrow id="id88285">
														<m:mi id="id88286">W</m:mi>
														<m:mo id="id88288" mathvariant="normal">:=</m:mo>
														<m:mrow id="id88293">
															<m:mrow id="id88294">
																<m:mrow id="id88295">
																	<m:mi id="id88296">W</m:mi>
																	<m:mo id="id88298" mathvariant="italic">⁢</m:mo>
																	<m:mfenced id="id88303" open="(" close=")">
																		<m:mi id="id88308">G</m:mi>
																	</m:mfenced>
																</m:mrow>
																<m:mo id="id88310" mathvariant="italic">/</m:mo>
																<m:mi id="id88314">W</m:mi>
															</m:mrow>
															<m:mo id="id88317" mathvariant="italic">⁢</m:mo>
															<m:mfenced id="id88321" open="(" close=")">
																<m:mi id="id88326">H</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id88328" encoding="MathML-Content">
														<m:apply id="id88332">
															<m:ci id="id88333">:=</m:ci>
															<m:ci id="id88335">W</m:ci>
															<m:apply id="id88337">
																<m:times id="id88338"/>
																<m:apply id="id88339">
																	<m:divide id="id88340"/>
																	<m:apply id="id88341">
																		<m:times id="id88342"/>
																		<m:ci id="id88344">W</m:ci>
																		<m:ci id="id88346">G</m:ci>
																	</m:apply>
																	<m:ci id="id88348">W</m:ci>
																</m:apply>
																<m:ci id="id88350">H</m:ci>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math> is the relative Weyl group.</span>
									</p>
								</div>

							</div>

						</div>

					</div>

					<div class="section" id="S7">
						<h2 id="id88365" class="title section-title">
							<span id="id88369" class="tag">7 </span>Example: The complex Grassmannian</h2>
						<div class="para" id="S7.p1">
							<p id="id88382" style="" class="p">This section contains a worked example: the complex Grassmannian of
								<m:math id="id88389" display="inline">
									<m:semantics id="id88392">
										<m:mi id="id88393">k</m:mi>
										<m:annotation-xml id="id88395" encoding="MathML-Content">
											<m:ci id="id88399">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-planes in <m:math id="id88402" display="inline">
									<m:semantics id="id88405">
										<m:msup id="id88406">
											<m:mi id="id88408" mathvariant="double-struck">C</m:mi>
											<m:mi id="id88412">n</m:mi>
										</m:msup>
										<m:annotation-xml id="id88414" encoding="MathML-Content">
											<m:apply id="id88417">
												<m:csymbol id="id88418" cd="ambiguous">superscript</m:csymbol>
												<m:ci id="id88423">C</m:ci>
												<m:ci id="id88425">n</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which we denote
								<m:math id="id88429" display="inline">
									<m:semantics id="id88432">
										<m:mrow id="id88433">
											<m:mi id="id88434">G</m:mi>
											<m:mo id="id88436">⁢</m:mo>
											<m:mfenced id="id88439" open="(" close=")">
												<m:mrow id="id88444">
													<m:mi id="id88445">k</m:mi>
													<m:mo id="id88447">,</m:mo>
													<m:mi id="id88449">n</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id88451" encoding="MathML-Content">
											<m:apply id="id88455">
												<m:times id="id88456"/>
												<m:ci id="id88457">G</m:ci>
												<m:apply id="id88459">
													<m:interval id="id88460" closure="open"/>
													<m:ci id="id88463">k</m:ci>
													<m:ci id="id88465">n</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. We first describe the results of
applying theorems A and B, and we then describe the derivation of
these results in more detail.</p>
						</div>

						<div class="para" id="S7.p2">
							<p id="id88480" style="" class="p">The Grassmannian can be described as the symplectic quotient of the
set of complex matrices with <m:math id="id88487" display="inline">
									<m:semantics id="id88490">
										<m:mi id="id88491">n</m:mi>
										<m:annotation-xml id="id88493" encoding="MathML-Content">
											<m:ci id="id88497">n</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> rows and <m:math id="id88500" display="inline">
									<m:semantics id="id88504">
										<m:mi id="id88505">k</m:mi>
										<m:annotation-xml id="id88507" encoding="MathML-Content">
											<m:ci id="id88510">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> columns by the unitary
group</p>
							<table class="equation" id="S7.Ex1">

								<tr valign="baseline" class="equation" id="S7.Ex1">
									<td id="id88530" class="eqpad"/>
									<td id="id88535" nowrap="yes" align="center" colspan="1">
										<m:math id="id88542" display="block">
											<m:semantics id="id88545">
												<m:mrow id="id88546">
													<m:mi id="id88548">G</m:mi>
													<m:mo id="id88550">(</m:mo>
													<m:mi id="id88552">k</m:mi>
													<m:mo id="id88554">,</m:mo>
													<m:mi id="id88556">n</m:mi>
													<m:mo id="id88558">)</m:mo>
													<m:mo id="id88560">≅</m:mo>
													<m:mo id="id88563">Hom</m:mo>
													<m:mo id="id88565">(</m:mo>
													<m:msup id="id88567">
														<m:mi id="id88568" mathvariant="double-struck">C</m:mi>
														<m:mi id="id88572">k</m:mi>
													</m:msup>
													<m:mo id="id88575">,</m:mo>
													<m:msup id="id88577">
														<m:mi id="id88578" mathvariant="double-struck">C</m:mi>
														<m:mi id="id88582">n</m:mi>
													</m:msup>
													<m:mo id="id88584">)</m:mo>
													<m:mo id="id88586">/</m:mo>
													<m:mo id="id88589">/</m:mo>
													<m:mi id="id88591">U</m:mi>
													<m:mo id="id88593">(</m:mo>
													<m:mi id="id88595">k</m:mi>
													<m:mo id="id88597">)</m:mo>
													<m:mo id="id88599">,</m:mo>
												</m:mrow>
												<m:annotation-xml id="id88601" encoding="MathML-Content"/>
											</m:semantics>
										</m:math>
									</td>
									<td id="id88606" class="eqpad"/>
								</tr>

							</table>
							<p id="id88612" style="" class="p">where <m:math id="id88619" display="inline">
									<m:semantics id="id88622">
										<m:mrow id="id88623">
											<m:mi id="id88624">g</m:mi>
											<m:mo id="id88626">∈</m:mo>
											<m:mrow id="id88629">
												<m:mi id="id88630">U</m:mi>
												<m:mo id="id88632">⁢</m:mo>
												<m:mfenced id="id88634" open="(" close=")">
													<m:mi id="id88639">k</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id88642" encoding="MathML-Content">
											<m:apply id="id88645">
												<m:ci id="id88646">∈</m:ci>
												<m:ci id="id88648">g</m:ci>
												<m:apply id="id88650">
													<m:times id="id88652"/>
													<m:ci id="id88653">U</m:ci>
													<m:ci id="id88655">k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> acts on a matrix <m:math id="id88658" display="inline">
									<m:semantics id="id88662">
										<m:mrow id="id88663">
											<m:mi id="id88664">A</m:mi>
											<m:mo id="id88666">∈</m:mo>
											<m:mrow id="id88669">
												<m:mo id="id88670">Hom</m:mo>
												<m:mo id="id88672">⁡</m:mo>
												<m:mfenced id="id88674" open="(" close=")">
													<m:mrow id="id88679">
														<m:msup id="id88680">
															<m:mi id="id88681" mathvariant="double-struck">C</m:mi>
															<m:mi id="id88686">k</m:mi>
														</m:msup>
														<m:mo id="id88688">,</m:mo>
														<m:msup id="id88690">
															<m:mi id="id88691" mathvariant="double-struck">C</m:mi>
															<m:mi id="id88696">n</m:mi>
														</m:msup>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id88698" encoding="MathML-Content">
											<m:apply id="id88701">
												<m:ci id="id88702">∈</m:ci>
												<m:ci id="id88704">A</m:ci>
												<m:apply id="id88707">
													<m:ci id="id88708">Hom</m:ci>
													<m:apply id="id88710">
														<m:csymbol id="id88711" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id88716">C</m:ci>
														<m:ci id="id88718">k</m:ci>
													</m:apply>
													<m:apply id="id88720">
														<m:csymbol id="id88721" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id88726">C</m:ci>
														<m:ci id="id88728">n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> by
								<m:math id="id88731" display="inline">
									<m:semantics id="id88734">
										<m:mrow id="id88736">
											<m:mi id="id88737">A</m:mi>
											<m:mo id="id88739">∘</m:mo>
											<m:msup id="id88741">
												<m:mi id="id88742">g</m:mi>
												<m:mrow id="id88744">
													<m:mo id="id88745">-</m:mo>
													<m:mn id="id88748">1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml id="id88750" encoding="MathML-Content">
											<m:apply id="id88753">
												<m:compose id="id88754"/>
												<m:ci id="id88755">A</m:ci>
												<m:apply id="id88757">
													<m:csymbol id="id88758" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id88763">g</m:ci>
													<m:apply id="id88765">
														<m:minus id="id88766"/>
														<m:cn id="id88767">1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.</p>
						</div>

						<div class="para" id="S7.p3">
							<p id="id88779" style="" class="p">The associated symplectic quotient by the maximal torus  <m:math id="id88786" display="inline">
									<m:semantics id="id88789">
										<m:mrow id="id88790">
											<m:mi id="id88792">T</m:mi>
											<m:mo id="id88794">⊂</m:mo>
											<m:mrow id="id88796">
												<m:mi id="id88797">U</m:mi>
												<m:mo id="id88799">⁢</m:mo>
												<m:mfenced id="id88802" open="(" close=")">
													<m:mi id="id88807">k</m:mi>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id88809" encoding="MathML-Content">
											<m:apply id="id88812">
												<m:ci id="id88813">⊂</m:ci>
												<m:ci id="id88816">T</m:ci>
												<m:apply id="id88818">
													<m:times id="id88819"/>
													<m:ci id="id88820">U</m:ci>
													<m:ci id="id88822">k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
turns out to be the <m:math id="id88826" display="inline">
									<m:semantics id="id88829">
										<m:mi id="id88830">k</m:mi>
										<m:annotation-xml id="id88832" encoding="MathML-Content">
											<m:ci id="id88835">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-fold product <m:math id="id88839" display="inline">
									<m:semantics id="id88842">
										<m:msup id="id88843">
											<m:mfenced id="id88844" open="(" close=")">
												<m:mrow id="id88849">
													<m:mi id="id88850" mathvariant="double-struck">C</m:mi>
													<m:mo id="id88855">⁢</m:mo>
													<m:msup id="id88857">
														<m:mi id="id88858" mathvariant="double-struck">P</m:mi>
														<m:mrow id="id88863">
															<m:mi id="id88864">n</m:mi>
															<m:mo id="id88866">-</m:mo>
															<m:mn id="id88868">1</m:mn>
														</m:mrow>
													</m:msup>
												</m:mrow>
											</m:mfenced>
											<m:mi id="id88870">k</m:mi>
										</m:msup>
										<m:annotation-xml id="id88872" encoding="MathML-Content">
											<m:apply id="id88876">
												<m:csymbol id="id88877" cd="ambiguous">superscript</m:csymbol>
												<m:apply id="id88881">
													<m:times id="id88882"/>
													<m:ci id="id88884">C</m:ci>
													<m:apply id="id88886">
														<m:csymbol id="id88887" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id88891">P</m:ci>
														<m:apply id="id88894">
															<m:minus id="id88895"/>
															<m:ci id="id88896">n</m:ci>
															<m:cn id="id88898">1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:ci id="id88900">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Its cohomology
ring  is generated by degree-<m:math id="id88904" display="inline">
									<m:semantics id="id88907">
										<m:mn id="id88908">2</m:mn>
										<m:annotation-xml id="id88910" encoding="MathML-Content">
											<m:cn id="id88914">2</m:cn>
										</m:annotation-xml>
									</m:semantics>
								</m:math> elements <m:math id="id88917" display="inline">
									<m:semantics id="id88920">
										<m:mfenced id="id88921" open="{" close="}">
											<m:mrow id="id88926">
												<m:msub id="id88928">
													<m:mi id="id88929">u</m:mi>
													<m:mn id="id88931">1</m:mn>
												</m:msub>
												<m:mo id="id88933">,</m:mo>
												<m:mi id="id88935" mathvariant="normal">…</m:mi>
												<m:mo id="id88940">,</m:mo>
												<m:msub id="id88942">
													<m:mi id="id88943">u</m:mi>
													<m:mi id="id88945">k</m:mi>
												</m:msub>
											</m:mrow>
										</m:mfenced>
										<m:annotation-xml id="id88947" encoding="MathML-Content">
											<m:apply id="id88950">
												<m:set id="id88952"/>
												<m:apply id="id88953">
													<m:csymbol id="id88954" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id88958">u</m:ci>
													<m:cn id="id88960">1</m:cn>
												</m:apply>
												<m:ci id="id88963">…</m:ci>
												<m:apply id="id88965">
													<m:csymbol id="id88966" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id88971">u</m:ci>
													<m:ci id="id88973">k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, where
								<m:math id="id88976" display="inline">
									<m:semantics id="id88980">
										<m:msub id="id88981">
											<m:mi id="id88982">u</m:mi>
											<m:mi id="id88984">i</m:mi>
										</m:msub>
										<m:annotation-xml id="id88986" encoding="MathML-Content">
											<m:apply id="id88989">
												<m:csymbol id="id88990" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id88995">u</m:ci>
												<m:ci id="id88997">i</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is the positive generator of the cohomology ring of the <m:math id="id89001" display="inline">
									<m:semantics id="id89004">
										<m:mi id="id89006">i</m:mi>
										<m:annotation-xml id="id89008" encoding="MathML-Content">
											<m:ci id="id89011">i</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-th
copy of <m:math id="id89014" display="inline">
									<m:semantics id="id89018">
										<m:mrow id="id89019">
											<m:mi id="id89020" mathvariant="double-struck">C</m:mi>
											<m:mo id="id89024">⁢</m:mo>
											<m:msup id="id89027">
												<m:mi id="id89028" mathvariant="double-struck">P</m:mi>
												<m:mrow id="id89032">
													<m:mi id="id89033">n</m:mi>
													<m:mo id="id89035">-</m:mo>
													<m:mn id="id89038">1</m:mn>
												</m:mrow>
											</m:msup>
										</m:mrow>
										<m:annotation-xml id="id89040" encoding="MathML-Content">
											<m:apply id="id89043">
												<m:times id="id89044"/>
												<m:ci id="id89045">C</m:ci>
												<m:apply id="id89047">
													<m:csymbol id="id89048" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id89053">P</m:ci>
													<m:apply id="id89055">
														<m:minus id="id89056"/>
														<m:ci id="id89057">n</m:ci>
														<m:cn id="id89059">1</m:cn>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. The Weyl group of <m:math id="id89063" display="inline">
									<m:semantics id="id89066">
										<m:mrow id="id89067">
											<m:mi id="id89068">U</m:mi>
											<m:mo id="id89070">⁢</m:mo>
											<m:mfenced id="id89073" open="(" close=")">
												<m:mi id="id89078">k</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id89080" encoding="MathML-Content">
											<m:apply id="id89083">
												<m:times id="id89084"/>
												<m:ci id="id89086">U</m:ci>
												<m:ci id="id89088">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is the
symmetric group on <m:math id="id89091" display="inline">
									<m:semantics id="id89095">
										<m:mi id="id89096">k</m:mi>
										<m:annotation-xml id="id89098" encoding="MathML-Content">
											<m:ci id="id89101">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> elements <m:math id="id89105" display="inline">
									<m:semantics id="id89108">
										<m:msup id="id89109">
											<m:mi id="id89110">S</m:mi>
											<m:mi id="id89112">k</m:mi>
										</m:msup>
										<m:annotation-xml id="id89114" encoding="MathML-Content">
											<m:apply id="id89118">
												<m:csymbol id="id89119" cd="ambiguous">superscript</m:csymbol>
												<m:ci id="id89123">S</m:ci>
												<m:ci id="id89126">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, which acts by permuting the
factors in <m:math id="id89129" display="inline">
									<m:semantics id="id89133">
										<m:msup id="id89134">
											<m:mfenced id="id89135" open="(" close=")">
												<m:mrow id="id89140">
													<m:mi id="id89141" mathvariant="double-struck">C</m:mi>
													<m:mo id="id89145">⁢</m:mo>
													<m:msup id="id89148">
														<m:mi id="id89149" mathvariant="double-struck">P</m:mi>
														<m:mrow id="id89153">
															<m:mi id="id89154">n</m:mi>
															<m:mo id="id89156">-</m:mo>
															<m:mn id="id89158">1</m:mn>
														</m:mrow>
													</m:msup>
												</m:mrow>
											</m:mfenced>
											<m:mi id="id89161">k</m:mi>
										</m:msup>
										<m:annotation-xml id="id89163" encoding="MathML-Content">
											<m:apply id="id89166">
												<m:csymbol id="id89167" cd="ambiguous">superscript</m:csymbol>
												<m:apply id="id89172">
													<m:times id="id89173"/>
													<m:ci id="id89174">C</m:ci>
													<m:apply id="id89176">
														<m:csymbol id="id89177" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id89182">P</m:ci>
														<m:apply id="id89184">
															<m:minus id="id89185"/>
															<m:ci id="id89186">n</m:ci>
															<m:cn id="id89188">1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
												<m:ci id="id89190">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. The roots <m:math id="id89194" display="inline">
									<m:semantics id="id89197">
										<m:mi id="id89198">α</m:mi>
										<m:annotation-xml id="id89201" encoding="MathML-Content">
											<m:ci id="id89204">α</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of <m:math id="id89208" display="inline">
									<m:semantics id="id89211">
										<m:mrow id="id89212">
											<m:mi id="id89213">U</m:mi>
											<m:mo id="id89215">⁢</m:mo>
											<m:mfenced id="id89218" open="(" close=")">
												<m:mi id="id89223">k</m:mi>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id89225" encoding="MathML-Content">
											<m:apply id="id89228">
												<m:times id="id89229"/>
												<m:ci id="id89230">U</m:ci>
												<m:ci id="id89232">k</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> can be enumerated by
pairs of positive integers <m:math id="id89236" display="inline">
									<m:semantics id="id89240">
										<m:mfenced id="id89241" open="(" close=")">
											<m:mrow id="id89246">
												<m:mi id="id89247">i</m:mi>
												<m:mo id="id89249">,</m:mo>
												<m:mi id="id89251">j</m:mi>
											</m:mrow>
										</m:mfenced>
										<m:annotation-xml id="id89253" encoding="MathML-Content">
											<m:apply id="id89256">
												<m:interval id="id89258" closure="open"/>
												<m:ci id="id89261">i</m:ci>
												<m:ci id="id89263">j</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> with <m:math id="id89266" display="inline">
									<m:semantics id="id89270">
										<m:mrow id="id89271">
											<m:mrow id="id89272">
												<m:mn id="id89273">1</m:mn>
												<m:mo id="id89275">≤</m:mo>
												<m:mi id="id89278">i</m:mi>
											</m:mrow>
											<m:mo id="id89280">,</m:mo>
											<m:mrow id="id89282">
												<m:mi id="id89283">j</m:mi>
												<m:mo id="id89285">≤</m:mo>
												<m:mi id="id89287">k</m:mi>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id89290" encoding="MathML-Content">
											<m:apply id="id89293">
												<m:ci id="id89294"/>
												<m:apply id="id89295">
													<m:leq id="id89296"/>
													<m:cn id="id89297">1</m:cn>
													<m:ci id="id89299">i</m:ci>
												</m:apply>
												<m:apply id="id89301">
													<m:leq id="id89302"/>
													<m:ci id="id89304">j</m:ci>
													<m:ci id="id89306">k</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> and <m:math id="id89309" display="inline">
									<m:semantics id="id89312">
										<m:mrow id="id89314">
											<m:mi id="id89315">i</m:mi>
											<m:mo id="id89317">≠</m:mo>
											<m:mi id="id89319">j</m:mi>
										</m:mrow>
										<m:annotation-xml id="id89321" encoding="MathML-Content">
											<m:apply id="id89325">
												<m:neq id="id89326"/>
												<m:ci id="id89327">i</m:ci>
												<m:ci id="id89329">j</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>,
and the cohomology class
corresponding to the root <m:math id="id89333" display="inline">
									<m:semantics id="id89336">
										<m:mfenced id="id89337" open="(" close=")">
											<m:mrow id="id89342">
												<m:mi id="id89344">i</m:mi>
												<m:mo id="id89346">,</m:mo>
												<m:mi id="id89348">j</m:mi>
											</m:mrow>
										</m:mfenced>
										<m:annotation-xml id="id89350" encoding="MathML-Content">
											<m:apply id="id89353">
												<m:interval id="id89354" closure="open"/>
												<m:ci id="id89358">i</m:ci>
												<m:ci id="id89360">j</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> is the class
								<m:math id="id89363" display="inline">
									<m:semantics id="id89367">
										<m:mrow id="id89368">
											<m:mrow id="id89369">
												<m:mi id="id89370">e</m:mi>
												<m:mo id="id89372">⁢</m:mo>
												<m:mfenced id="id89374" open="(" close=")">
													<m:mi id="id89379">α</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:mo id="id89382">=</m:mo>
											<m:mrow id="id89384">
												<m:msub id="id89385">
													<m:mi id="id89386">u</m:mi>
													<m:mi id="id89388">j</m:mi>
												</m:msub>
												<m:mo id="id89390">-</m:mo>
												<m:msub id="id89392">
													<m:mi id="id89394">u</m:mi>
													<m:mi id="id89396">i</m:mi>
												</m:msub>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id89398" encoding="MathML-Content">
											<m:apply id="id89401">
												<m:eq id="id89402"/>
												<m:apply id="id89403">
													<m:times id="id89404"/>
													<m:ci id="id89405">e</m:ci>
													<m:ci id="id89408">α</m:ci>
												</m:apply>
												<m:apply id="id89410">
													<m:minus id="id89411"/>
													<m:apply id="id89412">
														<m:csymbol id="id89413" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id89418">u</m:ci>
														<m:ci id="id89420">j</m:ci>
													</m:apply>
													<m:apply id="id89422">
														<m:csymbol id="id89423" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id89428">u</m:ci>
														<m:ci id="id89430">i</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. Hence, theorem A states</p>
						</div>

						<div class="theorem proposition" id="S7.E1">
							<h6 id="id89442" class="title theorem-title bold">
								<span id="id89446" class="tag">Proposition 7.1</span>.</h6>
							<div class="para" id="S7.E1.p1">
								<p id="id89459" style="" class="p">
									<span id="id89464" style="" class="text italic">The cohomology ring of the Grassmannian <m:math id="id89471" display="inline">
											<m:semantics id="id89474">
												<m:mrow id="id89476">
													<m:mi id="id89477">G</m:mi>
													<m:mo id="id89479" mathvariant="italic">⁢</m:mo>
													<m:mfenced id="id89483" open="(" close=")">
														<m:mrow id="id89488">
															<m:mi id="id89490">k</m:mi>
															<m:mo id="id89492" mathvariant="italic">,</m:mo>
															<m:mi id="id89496">n</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id89498" encoding="MathML-Content">
													<m:apply id="id89502">
														<m:times id="id89503"/>
														<m:ci id="id89504">G</m:ci>
														<m:apply id="id89506">
															<m:interval id="id89507" closure="open"/>
															<m:ci id="id89510">k</m:ci>
															<m:ci id="id89512">n</m:ci>
														</m:apply>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is given by
									</span>
								</p>
								<table class="equation" id="S7.Ex2">

									<tr valign="baseline" class="equation" id="S7.Ex2">
										<td id="id89532" class="eqpad"/>
										<td id="id89537" nowrap="yes" align="center" colspan="1">
											<m:math id="id89544" display="block">
												<m:semantics id="id89548">
													<m:mrow id="id89549">
														<m:mrow id="id89550">
															<m:mrow id="id89551">
																<m:msup id="id89552">
																	<m:mi id="id89553">H</m:mi>
																	<m:mo id="id89555">*</m:mo>
																</m:msup>
																<m:mo id="id89557">⁢</m:mo>
																<m:mfenced id="id89560" open="(" close=")">
																	<m:mrow id="id89565">
																		<m:mrow id="id89566">
																			<m:mi id="id89567">G</m:mi>
																			<m:mo id="id89569">⁢</m:mo>
																			<m:mfenced id="id89571" open="(" close=")">
																				<m:mrow id="id89576">
																					<m:mi id="id89577">k</m:mi>
																					<m:mo id="id89580">,</m:mo>
																					<m:mi id="id89582">n</m:mi>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																		<m:mo id="id89584">;</m:mo>
																		<m:mi id="id89586" mathvariant="double-struck">Q</m:mi>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id89590">≅</m:mo>
															<m:mfrac id="id89593">
																<m:mrow id="id89594">
																	<m:mi id="id89595" mathvariant="double-struck">Q</m:mi>
																	<m:mo id="id89599">⁢</m:mo>
																	<m:msup id="id89602">
																		<m:mfenced id="id89603" open="[" close="]">
																			<m:mrow id="id89608">
																				<m:msub id="id89609">
																					<m:mi id="id89610">u</m:mi>
																					<m:mn id="id89612">1</m:mn>
																				</m:msub>
																				<m:mo id="id89614">,</m:mo>
																				<m:mi id="id89616" mathvariant="normal">…</m:mi>
																				<m:mo id="id89621">,</m:mo>
																				<m:msub id="id89623">
																					<m:mi id="id89624">u</m:mi>
																					<m:mi id="id89626">k</m:mi>
																				</m:msub>
																			</m:mrow>
																		</m:mfenced>
																		<m:msub id="id89628">
																			<m:mi id="id89630">S</m:mi>
																			<m:mi id="id89632">k</m:mi>
																		</m:msub>
																	</m:msup>
																</m:mrow>
																<m:mrow id="id89634">
																	<m:mfenced id="id89637" open="⟨" close="⟩">
																		<m:mrow id="id89642">
																			<m:msubsup id="id89643">
																				<m:mi id="id89644">u</m:mi>
																				<m:mn id="id89647">1</m:mn>
																				<m:mi id="id89649">n</m:mi>
																			</m:msubsup>
																			<m:mo id="id89651">,</m:mo>
																			<m:mi id="id89653" mathvariant="normal">…</m:mi>
																			<m:mo id="id89657">,</m:mo>
																			<m:msubsup id="id89659">
																				<m:mi id="id89660">u</m:mi>
																				<m:mi id="id89662">k</m:mi>
																				<m:mi id="id89664">n</m:mi>
																			</m:msubsup>
																		</m:mrow>
																	</m:mfenced>
																	<m:mo id="id89666">:</m:mo>
																	<m:mrow id="id89668">
																		<m:msub id="id89670">
																			<m:mo id="id89671">∏</m:mo>
																			<m:mrow id="id89673">
																				<m:mi id="id89674">i</m:mi>
																				<m:mo id="id89676">≠</m:mo>
																				<m:mi id="id89679">j</m:mi>
																			</m:mrow>
																		</m:msub>
																		<m:mfenced id="id89681" open="(" close=")">
																			<m:mrow id="id89685">
																				<m:msub id="id89686">
																					<m:mi id="id89687">u</m:mi>
																					<m:mi id="id89689">i</m:mi>
																				</m:msub>
																				<m:mo id="id89691">-</m:mo>
																				<m:msub id="id89693">
																					<m:mi id="id89694">u</m:mi>
																					<m:mi id="id89697">j</m:mi>
																				</m:msub>
																			</m:mrow>
																		</m:mfenced>
																	</m:mrow>
																</m:mrow>
															</m:mfrac>
														</m:mrow>
														<m:mo id="id89699">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id89701" encoding="MathML-Content">
														<m:apply id="id89704">
															<m:approx id="id89705"/>
															<m:apply id="id89706">
																<m:times id="id89707"/>
																<m:apply id="id89708">
																	<m:csymbol id="id89710" cd="ambiguous">superscript</m:csymbol>
																	<m:ci id="id89714">H</m:ci>
																	<m:times id="id89716"/>
																</m:apply>
																<m:apply id="id89717">
																	<m:list id="id89718"/>
																	<m:apply id="id89720">
																		<m:times id="id89721"/>
																		<m:ci id="id89722">G</m:ci>
																		<m:apply id="id89724">
																			<m:interval id="id89725" closure="open"/>
																			<m:ci id="id89728">k</m:ci>
																			<m:ci id="id89730">n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:ci id="id89732">Q</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id89735">
																<m:divide id="id89736"/>
																<m:apply id="id89737">
																	<m:times id="id89738"/>
																	<m:ci id="id89739">Q</m:ci>
																	<m:apply id="id89741">
																		<m:csymbol id="id89742" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id89747">
																			<m:list id="id89748"/>
																			<m:apply id="id89749">
																				<m:csymbol id="id89750" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89755">u</m:ci>
																				<m:cn id="id89757">1</m:cn>
																			</m:apply>
																			<m:ci id="id89759">…</m:ci>
																			<m:apply id="id89761">
																				<m:csymbol id="id89762" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89767">u</m:ci>
																				<m:ci id="id89769">k</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id89771">
																			<m:csymbol id="id89772" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id89777">S</m:ci>
																			<m:ci id="id89779">k</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id89781">
																	<m:ci id="id89782">:</m:ci>
																	<m:apply id="id89784">
																		<m:list id="id89786"/>
																		<m:apply id="id89787">
																			<m:csymbol id="id89788" cd="ambiguous">superscript</m:csymbol>
																			<m:apply id="id89792">
																				<m:csymbol id="id89793" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89798">u</m:ci>
																				<m:cn id="id89800">1</m:cn>
																			</m:apply>
																			<m:ci id="id89802">n</m:ci>
																		</m:apply>
																		<m:ci id="id89804">…</m:ci>
																		<m:apply id="id89807">
																			<m:csymbol id="id89808" cd="ambiguous">superscript</m:csymbol>
																			<m:apply id="id89813">
																				<m:csymbol id="id89814" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89818">u</m:ci>
																				<m:ci id="id89820">k</m:ci>
																			</m:apply>
																			<m:ci id="id89823">n</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id89825">
																		<m:apply id="id89826">
																			<m:csymbol id="id89827" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id89832">∏</m:ci>
																			<m:apply id="id89834">
																				<m:neq id="id89835"/>
																				<m:ci id="id89836">i</m:ci>
																				<m:ci id="id89838">j</m:ci>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id89840">
																			<m:minus id="id89841"/>
																			<m:apply id="id89842">
																				<m:csymbol id="id89844" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89848">u</m:ci>
																				<m:ci id="id89850">i</m:ci>
																			</m:apply>
																			<m:apply id="id89852">
																				<m:csymbol id="id89854" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id89858">u</m:ci>
																				<m:ci id="id89860">j</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id89864" class="eqpad"/>
									</tr>

								</table>
								<p id="id89870" style="" class="p">
									<span id="id89875" style="" class="text italic">where the expression <m:math id="id89882" display="inline">
											<m:semantics id="id89885">
												<m:mrow id="id89886">
													<m:mi id="id89887">I</m:mi>
													<m:mo id="id89889" mathvariant="normal">:</m:mo>
													<m:mi id="id89894">e</m:mi>
												</m:mrow>
												<m:annotation-xml id="id89896" encoding="MathML-Content">
													<m:apply id="id89899">
														<m:ci id="id89900">:</m:ci>
														<m:ci id="id89902">I</m:ci>
														<m:ci id="id89905">e</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> in the denominator denotes the
ideal quotient of the ideal <m:math id="id89908" display="inline">
											<m:semantics id="id89912">
												<m:mi id="id89913">I</m:mi>
												<m:annotation-xml id="id89915" encoding="MathML-Content">
													<m:ci id="id89918">I</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> by the element <m:math id="id89922" display="inline">
											<m:semantics id="id89925">
												<m:mi id="id89926">e</m:mi>
												<m:annotation-xml id="id89928" encoding="MathML-Content">
													<m:ci id="id89932">e</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, that is,
the ideal consisting of all elements <m:math id="id89935" display="inline">
											<m:semantics id="id89939">
												<m:mi id="id89940">b</m:mi>
												<m:annotation-xml id="id89942" encoding="MathML-Content">
													<m:ci id="id89945">b</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> such that <m:math id="id89949" display="inline">
											<m:semantics id="id89952">
												<m:mrow id="id89953">
													<m:mrow id="id89954">
														<m:mi id="id89955">b</m:mi>
														<m:mo id="id89957" mathvariant="normal">⋅</m:mo>
														<m:mi id="id89962">e</m:mi>
													</m:mrow>
													<m:mo id="id89964" mathvariant="normal">∈</m:mo>
													<m:mi id="id89969">I</m:mi>
												</m:mrow>
												<m:annotation-xml id="id89971" encoding="MathML-Content">
													<m:apply id="id89974">
														<m:ci id="id89975">∈</m:ci>
														<m:apply id="id89978">
															<m:ci id="id89979">⋅</m:ci>
															<m:ci id="id89981">b</m:ci>
															<m:ci id="id89983">e</m:ci>
														</m:apply>
														<m:ci id="id89986">I</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</span>
								</p>
							</div>

						</div>

						<div class="para" id="S7.p4">
							<p id="id89999" style="" class="p">The Grassmannian possesses a tautological vector bundle <m:math id="id90005" display="inline">
									<m:semantics id="id90009">
										<m:mrow id="id90010">
											<m:mi id="id90011">V</m:mi>
											<m:mo id="id90013">→</m:mo>
											<m:mrow id="id90015">
												<m:mi id="id90016">G</m:mi>
												<m:mo id="id90019">⁢</m:mo>
												<m:mfenced id="id90021" open="(" close=")">
													<m:mrow id="id90026">
														<m:mi id="id90027">k</m:mi>
														<m:mo id="id90029">,</m:mo>
														<m:mi id="id90031">n</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
										</m:mrow>
										<m:annotation-xml id="id90034" encoding="MathML-Content">
											<m:apply id="id90037">
												<m:ci id="id90038">→</m:ci>
												<m:ci id="id90040">V</m:ci>
												<m:apply id="id90042">
													<m:times id="id90044"/>
													<m:ci id="id90045">G</m:ci>
													<m:apply id="id90047">
														<m:interval id="id90048" closure="open"/>
														<m:ci id="id90051">k</m:ci>
														<m:ci id="id90053">n</m:ci>
													</m:apply>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math> of rank <m:math id="id90057" display="inline">
									<m:semantics id="id90060">
										<m:mi id="id90061">k</m:mi>
										<m:annotation-xml id="id90063" encoding="MathML-Content">
											<m:ci id="id90067">k</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>, and the cohomology ring is generated by the Chern
classes of the dual bundle <m:math id="id90071" display="inline">
									<m:semantics id="id90074">
										<m:msup id="id90075">
											<m:mi id="id90076">V</m:mi>
											<m:mo id="id90078">*</m:mo>
										</m:msup>
										<m:annotation-xml id="id90080" encoding="MathML-Content">
											<m:apply id="id90084">
												<m:csymbol id="id90085" cd="ambiguous">superscript</m:csymbol>
												<m:ci id="id90089">V</m:ci>
												<m:times id="id90092"/>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>. In the above description, <m:math id="id90094" display="inline">
									<m:semantics id="id90097">
										<m:mrow id="id90098">
											<m:msub id="id90100">
												<m:mi id="id90101">c</m:mi>
												<m:mi id="id90103">i</m:mi>
											</m:msub>
											<m:mo id="id90105">⁢</m:mo>
											<m:mfenced id="id90107" open="(" close=")">
												<m:msup id="id90112">
													<m:mi id="id90113">V</m:mi>
													<m:mo id="id90116">*</m:mo>
												</m:msup>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id90118" encoding="MathML-Content">
											<m:apply id="id90121">
												<m:times id="id90122"/>
												<m:apply id="id90123">
													<m:csymbol id="id90124" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id90129">c</m:ci>
													<m:ci id="id90131">i</m:ci>
												</m:apply>
												<m:apply id="id90133">
													<m:csymbol id="id90134" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id90139">V</m:ci>
													<m:times id="id90141"/>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
is represented by the <m:math id="id90143" display="inline">
									<m:semantics id="id90147">
										<m:mi id="id90148">i</m:mi>
										<m:annotation-xml id="id90150" encoding="MathML-Content">
											<m:ci id="id90153">i</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>-th symmetric polynomial in the <m:math id="id90157" display="inline">
									<m:semantics id="id90160">
										<m:msub id="id90161">
											<m:mi id="id90162">u</m:mi>
											<m:mi id="id90164">j</m:mi>
										</m:msub>
										<m:annotation-xml id="id90167" encoding="MathML-Content">
											<m:apply id="id90170">
												<m:csymbol id="id90171" cd="ambiguous">subscript</m:csymbol>
												<m:ci id="id90176">u</m:ci>
												<m:ci id="id90178">j</m:ci>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>.
The above description of the cohomology ring is quite different from the
usual description (which involves the Segre classes of <m:math id="id90182" display="inline">
									<m:semantics id="id90186">
										<m:mi id="id90187">V</m:mi>
										<m:annotation-xml id="id90189" encoding="MathML-Content">
											<m:ci id="id90192">V</m:ci>
										</m:annotation-xml>
									</m:semantics>
								</m:math>), and I have been
able to find neither a general algebraic proof of the equivalence of
the two descriptions, nor any reference to the above description in
the literature.</p>
						</div>

						<div class="para" id="S7.p5">
							<p id="id90206" style="" class="p">Theorem B gives</p>
						</div>

						<div class="theorem proposition" id="S7.E2">
							<h6 id="id90221" class="title theorem-title bold">
								<span id="id90224" class="tag">Proposition 7.2</span>.</h6>
							<div class="para" id="S7.E2.p1">
								<table class="equation" id="S7.Ex3">

									<tr valign="baseline" class="equation" id="S7.Ex3">
										<td id="id90252" class="eqpad"/>
										<td id="id90257" nowrap="yes" align="center" colspan="1">
											<m:math id="id90264" display="block">
												<m:semantics id="id90267">
													<m:mrow id="id90268">
														<m:mrow id="id90269">
															<m:msub id="id90270">
																<m:mo id="id90272">∫</m:mo>
																<m:mrow id="id90274">
																	<m:mi id="id90275">G</m:mi>
																	<m:mo id="id90277">⁢</m:mo>
																	<m:mfenced id="id90280" open="(" close=")">
																		<m:mrow id="id90285">
																			<m:mi id="id90286">k</m:mi>
																			<m:mo id="id90288">,</m:mo>
																			<m:mi id="id90290">n</m:mi>
																		</m:mrow>
																	</m:mfenced>
																</m:mrow>
															</m:msub>
															<m:msub id="id90292">
																<m:mi id="id90293">c</m:mi>
																<m:mn id="id90295">1</m:mn>
															</m:msub>
															<m:mo id="id90297">⁢</m:mo>
															<m:msup id="id90300">
																<m:mfenced id="id90301" open="(" close=")">
																	<m:msup id="id90306">
																		<m:mi id="id90307">V</m:mi>
																		<m:mo id="id90309">*</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:msub id="id90311">
																	<m:mi id="id90312">m</m:mi>
																	<m:mn id="id90314">1</m:mn>
																</m:msub>
															</m:msup>
														</m:mrow>
														<m:mo id="id90317">⌣</m:mo>
														<m:mi id="id90319" mathvariant="normal">…</m:mi>
														<m:mo id="id90324">⌣</m:mo>
														<m:mrow id="id90326">
															<m:msub id="id90327">
																<m:mi id="id90328">c</m:mi>
																<m:mi id="id90330">k</m:mi>
															</m:msub>
															<m:mo id="id90332">⁢</m:mo>
															<m:msup id="id90335">
																<m:mfenced id="id90336" open="(" close=")">
																	<m:msup id="id90341">
																		<m:mi id="id90342">V</m:mi>
																		<m:mo id="id90344">*</m:mo>
																	</m:msup>
																</m:mfenced>
																<m:msub id="id90346">
																	<m:mi id="id90347">m</m:mi>
																	<m:mi id="id90350">k</m:mi>
																</m:msub>
															</m:msup>
														</m:mrow>
														<m:mo id="id90352">=</m:mo>
														<m:mrow id="id90354">
															<m:mfrac id="id90355">
																<m:mn id="id90356">1</m:mn>
																<m:mrow id="id90358">
																	<m:mi id="id90359">k</m:mi>
																	<m:mi id="id90361" mathvariant="normal">!</m:mi>
																</m:mrow>
															</m:mfrac>
															<m:mo id="id90366">⁢</m:mo>
															<m:mrow id="id90368">
																<m:msub id="id90369">
																	<m:mo id="id90370">coeff</m:mo>
																	<m:mrow id="id90373">
																		<m:msubsup id="id90374">
																			<m:mi id="id90375">u</m:mi>
																			<m:mn id="id90377">1</m:mn>
																			<m:mrow id="id90379">
																				<m:mi id="id90380">n</m:mi>
																				<m:mo id="id90382">-</m:mo>
																				<m:mn id="id90384">1</m:mn>
																			</m:mrow>
																		</m:msubsup>
																		<m:mo id="id90386">⁢</m:mo>
																		<m:mi id="id90389" mathvariant="normal">…</m:mi>
																		<m:mo id="id90394">⁢</m:mo>
																		<m:msubsup id="id90396">
																			<m:mi id="id90397">u</m:mi>
																			<m:mi id="id90399">k</m:mi>
																			<m:mrow id="id90401">
																				<m:mi id="id90402">n</m:mi>
																				<m:mo id="id90404">-</m:mo>
																				<m:mn id="id90407">1</m:mn>
																			</m:mrow>
																		</m:msubsup>
																	</m:mrow>
																</m:msub>
																<m:mo id="id90409">⁡</m:mo>
																<m:mfenced id="id90411" open="(" close=")">
																	<m:mrow id="id90416">
																		<m:mrow id="id90417">
																			<m:msubsup id="id90418">
																				<m:mi id="id90419">σ</m:mi>
																				<m:mn id="id90422">1</m:mn>
																				<m:msub id="id90424">
																					<m:mi id="id90425">m</m:mi>
																					<m:mn id="id90427">1</m:mn>
																				</m:msub>
																			</m:msubsup>
																			<m:mo id="id90429">⁢</m:mo>
																			<m:mi id="id90432" mathvariant="normal">…</m:mi>
																			<m:mo id="id90436">⁢</m:mo>
																			<m:msubsup id="id90439">
																				<m:mi id="id90440">σ</m:mi>
																				<m:mi id="id90442">k</m:mi>
																				<m:msub id="id90444">
																					<m:mi id="id90445">m</m:mi>
																					<m:mi id="id90448">k</m:mi>
																				</m:msub>
																			</m:msubsup>
																		</m:mrow>
																		<m:mo id="id90450">⋅</m:mo>
																		<m:mrow id="id90452">
																			<m:munder id="id90453">
																				<m:mo id="id90454" movablelimits="false">∏</m:mo>
																				<m:mrow id="id90459">
																					<m:mi id="id90460">i</m:mi>
																					<m:mo id="id90462">≠</m:mo>
																					<m:mi id="id90464">j</m:mi>
																				</m:mrow>
																			</m:munder>
																			<m:mfenced id="id90467" open="(" close=")">
																				<m:mrow id="id90472">
																					<m:msub id="id90473">
																						<m:mi id="id90474">u</m:mi>
																						<m:mi id="id90476">i</m:mi>
																					</m:msub>
																					<m:mo id="id90478">-</m:mo>
																					<m:msub id="id90480">
																						<m:mi id="id90481">u</m:mi>
																						<m:mi id="id90483">j</m:mi>
																					</m:msub>
																				</m:mrow>
																			</m:mfenced>
																		</m:mrow>
																	</m:mrow>
																</m:mfenced>
															</m:mrow>
														</m:mrow>
													</m:mrow>
													<m:annotation-xml id="id90486" encoding="MathML-Content">
														<m:apply id="id90489">
															<m:ci id="id90490"/>
															<m:apply id="id90491">
																<m:apply id="id90492">
																	<m:csymbol id="id90493" cd="ambiguous">subscript</m:csymbol>
																	<m:int id="id90498"/>
																	<m:apply id="id90499">
																		<m:times id="id90500"/>
																		<m:ci id="id90501">G</m:ci>
																		<m:apply id="id90503">
																			<m:interval id="id90504" closure="open"/>
																			<m:ci id="id90508">k</m:ci>
																			<m:ci id="id90510">n</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id90512">
																	<m:times id="id90513"/>
																	<m:apply id="id90514">
																		<m:csymbol id="id90515" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id90520">c</m:ci>
																		<m:cn id="id90522">1</m:cn>
																	</m:apply>
																	<m:apply id="id90524">
																		<m:csymbol id="id90525" cd="ambiguous">superscript</m:csymbol>
																		<m:apply id="id90530">
																			<m:csymbol id="id90531" cd="ambiguous">superscript</m:csymbol>
																			<m:ci id="id90535">V</m:ci>
																			<m:times id="id90538"/>
																		</m:apply>
																		<m:apply id="id90539">
																			<m:csymbol id="id90540" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id90544">m</m:ci>
																			<m:cn id="id90546">1</m:cn>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:ci id="id90549">⌣</m:ci>
															<m:ci id="id90551">…</m:ci>
															<m:ci id="id90553">⌣</m:ci>
															<m:apply id="id90556">
																<m:times id="id90557"/>
																<m:apply id="id90558">
																	<m:csymbol id="id90559" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id90564">c</m:ci>
																	<m:ci id="id90566">k</m:ci>
																</m:apply>
																<m:apply id="id90568">
																	<m:csymbol id="id90569" cd="ambiguous">superscript</m:csymbol>
																	<m:apply id="id90574">
																		<m:csymbol id="id90575" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id90579">V</m:ci>
																		<m:times id="id90582"/>
																	</m:apply>
																	<m:apply id="id90583">
																		<m:csymbol id="id90584" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id90588">m</m:ci>
																		<m:ci id="id90590">k</m:ci>
																	</m:apply>
																</m:apply>
															</m:apply>
															<m:eq id="id90593"/>
															<m:apply id="id90594">
																<m:times id="id90595"/>
																<m:apply id="id90596">
																	<m:divide id="id90597"/>
																	<m:cn id="id90598">1</m:cn>
																	<m:apply id="id90600">
																		<m:factorial id="id90601"/>
																		<m:ci id="id90602">k</m:ci>
																	</m:apply>
																</m:apply>
																<m:apply id="id90604">
																	<m:apply id="id90605">
																		<m:csymbol id="id90606" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id90611">coeff</m:ci>
																		<m:apply id="id90614">
																			<m:times id="id90615"/>
																			<m:apply id="id90616">
																				<m:csymbol id="id90617" cd="ambiguous">superscript</m:csymbol>
																				<m:apply id="id90621">
																					<m:csymbol id="id90622" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90627">u</m:ci>
																					<m:cn id="id90629">1</m:cn>
																				</m:apply>
																				<m:apply id="id90631">
																					<m:minus id="id90632"/>
																					<m:ci id="id90634">n</m:ci>
																					<m:cn id="id90636">1</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id90638">…</m:ci>
																			<m:apply id="id90640">
																				<m:csymbol id="id90641" cd="ambiguous">superscript</m:csymbol>
																				<m:apply id="id90646">
																					<m:csymbol id="id90647" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90652">u</m:ci>
																					<m:ci id="id90654">k</m:ci>
																				</m:apply>
																				<m:apply id="id90656">
																					<m:minus id="id90657"/>
																					<m:ci id="id90658">n</m:ci>
																					<m:cn id="id90660">1</m:cn>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id90662">
																		<m:ci id="id90663">⋅</m:ci>
																		<m:apply id="id90666">
																			<m:times id="id90667"/>
																			<m:apply id="id90668">
																				<m:csymbol id="id90669" cd="ambiguous">superscript</m:csymbol>
																				<m:apply id="id90674">
																					<m:csymbol id="id90675" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90679">σ</m:ci>
																					<m:cn id="id90682">1</m:cn>
																				</m:apply>
																				<m:apply id="id90684">
																					<m:csymbol id="id90685" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90690">m</m:ci>
																					<m:cn id="id90692">1</m:cn>
																				</m:apply>
																			</m:apply>
																			<m:ci id="id90694">…</m:ci>
																			<m:apply id="id90696">
																				<m:csymbol id="id90697" cd="ambiguous">superscript</m:csymbol>
																				<m:apply id="id90702">
																					<m:csymbol id="id90703" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90708">σ</m:ci>
																					<m:ci id="id90710">k</m:ci>
																				</m:apply>
																				<m:apply id="id90712">
																					<m:csymbol id="id90713" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90718">m</m:ci>
																					<m:ci id="id90720">k</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																		<m:apply id="id90722">
																			<m:apply id="id90723">
																				<m:csymbol id="id90724" cd="ambiguous">subscript</m:csymbol>
																				<m:ci id="id90729">∏</m:ci>
																				<m:apply id="id90732">
																					<m:neq id="id90733"/>
																					<m:ci id="id90734">i</m:ci>
																					<m:ci id="id90736">j</m:ci>
																				</m:apply>
																			</m:apply>
																			<m:apply id="id90738">
																				<m:minus id="id90739"/>
																				<m:apply id="id90740">
																					<m:csymbol id="id90741" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90746">u</m:ci>
																					<m:ci id="id90748">i</m:ci>
																				</m:apply>
																				<m:apply id="id90750">
																					<m:csymbol id="id90751" cd="ambiguous">subscript</m:csymbol>
																					<m:ci id="id90756">u</m:ci>
																					<m:ci id="id90758">j</m:ci>
																				</m:apply>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id90761" class="eqpad"/>
									</tr>

								</table>
								<p id="id90768" style="" class="p">
									<span id="id90773" style="" class="text italic">where <m:math id="id90779" display="inline">
											<m:semantics id="id90783">
												<m:msub id="id90784">
													<m:mi id="id90785">σ</m:mi>
													<m:mi id="id90787">i</m:mi>
												</m:msub>
												<m:annotation-xml id="id90789" encoding="MathML-Content">
													<m:apply id="id90793">
														<m:csymbol id="id90794" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id90798">σ</m:ci>
														<m:ci id="id90801">i</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> is the <m:math id="id90804" display="inline">
											<m:semantics id="id90808">
												<m:mi id="id90809">i</m:mi>
												<m:annotation-xml id="id90811" encoding="MathML-Content">
													<m:ci id="id90814">i</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>-th elementary symmetric polynomial of the
										<m:math id="id90818" display="inline">
											<m:semantics id="id90821">
												<m:msub id="id90822">
													<m:mi id="id90823">u</m:mi>
													<m:mi id="id90826">j</m:mi>
												</m:msub>
												<m:annotation-xml id="id90828" encoding="MathML-Content">
													<m:apply id="id90831">
														<m:csymbol id="id90832" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id90837">u</m:ci>
														<m:ci id="id90839">j</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math>, and <m:math id="id90842" display="inline">
											<m:semantics id="id90846">
												<m:mrow id="id90847">
													<m:msub id="id90848">
														<m:mo id="id90849" mathvariant="normal">coeff</m:mo>
														<m:mi id="id90854">m</m:mi>
													</m:msub>
													<m:mo id="id90856" mathvariant="italic">⁡</m:mo>
													<m:mfenced id="id90860" open="(" close=")">
														<m:mi id="id90865">p</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:annotation-xml id="id90868" encoding="MathML-Content">
													<m:apply id="id90871">
														<m:apply id="id90872">
															<m:csymbol id="id90873" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id90878">coeff</m:ci>
															<m:ci id="id90880">m</m:ci>
														</m:apply>
														<m:ci id="id90882">p</m:ci>
													</m:apply>
												</m:annotation-xml>
											</m:semantics>
										</m:math> denotes the coefficient of the
monomial <m:math id="id90886" display="inline">
											<m:semantics id="id90889">
												<m:mi id="id90890">m</m:mi>
												<m:annotation-xml id="id90892" encoding="MathML-Content">
													<m:ci id="id90896">m</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math> in the polynomial <m:math id="id90899" display="inline">
											<m:semantics id="id90903">
												<m:mi id="id90904">p</m:mi>
												<m:annotation-xml id="id90906" encoding="MathML-Content">
													<m:ci id="id90909">p</m:ci>
												</m:annotation-xml>
											</m:semantics>
										</m:math>.</span>
								</p>
							</div>

						</div>

						<div class="subsection" id="S7.SS1">
							<h3 id="id90922" class="title subsection-title">
								<span id="id90926" class="tag">7.1 </span>The construction of <m:math id="id90931" display="inline">
									<m:semantics id="id90935">
										<m:mrow id="id90936">
											<m:mi id="id90937">G</m:mi>
											<m:mo id="id90939">⁢</m:mo>
											<m:mfenced id="id90941" open="(" close=")">
												<m:mrow id="id90946">
													<m:mi id="id90947">k</m:mi>
													<m:mo id="id90950">,</m:mo>
													<m:mi id="id90952">n</m:mi>
												</m:mrow>
											</m:mfenced>
										</m:mrow>
										<m:annotation-xml id="id90954" encoding="MathML-Content">
											<m:apply id="id90957">
												<m:times id="id90958"/>
												<m:ci id="id90959">G</m:ci>
												<m:apply id="id90961">
													<m:interval id="id90962" closure="open"/>
													<m:ci id="id90966">k</m:ci>
													<m:ci id="id90968">n</m:ci>
												</m:apply>
											</m:apply>
										</m:annotation-xml>
									</m:semantics>
								</m:math>
							</h3>
							<div class="para" id="S7.SS1.p1">
								<p id="id90978" style="" class="p">The symplectic structure on <m:math id="id90984" display="inline">
										<m:semantics id="id90988">
											<m:mrow id="id90989">
												<m:mo id="id90990">Hom</m:mo>
												<m:mo id="id90992">⁡</m:mo>
												<m:mfenced id="id90994" open="(" close=")">
													<m:mrow id="id90999">
														<m:msup id="id91000">
															<m:mi id="id91002" mathvariant="double-struck">C</m:mi>
															<m:mi id="id91006">k</m:mi>
														</m:msup>
														<m:mo id="id91008">,</m:mo>
														<m:msup id="id91010">
															<m:mi id="id91011" mathvariant="double-struck">C</m:mi>
															<m:mi id="id91016">n</m:mi>
														</m:msup>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id91018" encoding="MathML-Content">
												<m:apply id="id91021">
													<m:ci id="id91022">Hom</m:ci>
													<m:apply id="id91024">
														<m:csymbol id="id91025" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id91030">C</m:ci>
														<m:ci id="id91032">k</m:ci>
													</m:apply>
													<m:apply id="id91034">
														<m:csymbol id="id91035" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id91040">C</m:ci>
														<m:ci id="id91042">n</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is the standard one for
a complex vector space with coordinates, namely, if
									<m:math id="id91046" display="inline">
										<m:semantics id="id91050">
											<m:mrow id="id91051">
												<m:msub id="id91052">
													<m:mi id="id91053">a</m:mi>
													<m:mrow id="id91055">
														<m:mi id="id91056">i</m:mi>
														<m:mo id="id91058">⁢</m:mo>
														<m:mi id="id91060">j</m:mi>
													</m:mrow>
												</m:msub>
												<m:mo id="id91063">=</m:mo>
												<m:mrow id="id91065">
													<m:msub id="id91066">
														<m:mi id="id91067">x</m:mi>
														<m:mrow id="id91069">
															<m:mi id="id91070">i</m:mi>
															<m:mo id="id91072">⁢</m:mo>
															<m:mi id="id91075">j</m:mi>
														</m:mrow>
													</m:msub>
													<m:mo id="id91077">+</m:mo>
													<m:mrow id="id91079">
														<m:msqrt id="id91080">
															<m:mrow id="id91081">
																<m:mo id="id91082">-</m:mo>
																<m:mn id="id91084">1</m:mn>
															</m:mrow>
														</m:msqrt>
														<m:mo id="id91086">⁢</m:mo>
														<m:msub id="id91089">
															<m:mi id="id91090">y</m:mi>
															<m:mrow id="id91092">
																<m:mi id="id91093">i</m:mi>
																<m:mo id="id91095">⁢</m:mo>
																<m:mi id="id91098">j</m:mi>
															</m:mrow>
														</m:msub>
													</m:mrow>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id91100" encoding="MathML-Content">
												<m:apply id="id91103">
													<m:eq id="id91104"/>
													<m:apply id="id91105">
														<m:csymbol id="id91106" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id91111">a</m:ci>
														<m:apply id="id91113">
															<m:times id="id91114"/>
															<m:ci id="id91115">i</m:ci>
															<m:ci id="id91117">j</m:ci>
														</m:apply>
													</m:apply>
													<m:apply id="id91119">
														<m:plus id="id91120"/>
														<m:apply id="id91122">
															<m:csymbol id="id91123" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id91127">x</m:ci>
															<m:apply id="id91129">
																<m:times id="id91130"/>
																<m:ci id="id91132">i</m:ci>
																<m:ci id="id91134">j</m:ci>
															</m:apply>
														</m:apply>
														<m:apply id="id91136">
															<m:times id="id91137"/>
															<m:apply id="id91138">
																<m:ci id="id91139"/>
																<m:apply id="id91140">
																	<m:minus id="id91141"/>
																	<m:cn id="id91142">1</m:cn>
																</m:apply>
															</m:apply>
															<m:apply id="id91144">
																<m:csymbol id="id91145" cd="ambiguous">subscript</m:csymbol>
																<m:ci id="id91150">y</m:ci>
																<m:apply id="id91152">
																	<m:times id="id91153"/>
																	<m:ci id="id91154">i</m:ci>
																	<m:ci id="id91156">j</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, for <m:math id="id91160" display="inline">
										<m:semantics id="id91163">
											<m:mrow id="id91164">
												<m:mrow id="id91165">
													<m:mn id="id91166">1</m:mn>
													<m:mo id="id91169">≤</m:mo>
													<m:mi id="id91171">i</m:mi>
													<m:mo id="id91173">≤</m:mo>
													<m:mi id="id91176">n</m:mi>
												</m:mrow>
												<m:mo id="id91178">,</m:mo>
												<m:mrow id="id91180">
													<m:mn id="id91181">1</m:mn>
													<m:mo id="id91183">≤</m:mo>
													<m:mi id="id91185">j</m:mi>
													<m:mo id="id91188">≤</m:mo>
													<m:mi id="id91190">k</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id91192" encoding="MathML-Content">
												<m:apply id="id91195">
													<m:ci id="id91196"/>
													<m:apply id="id91198">
														<m:ci id="id91199"/>
														<m:cn id="id91200">1</m:cn>
														<m:leq id="id91202"/>
														<m:ci id="id91203">i</m:ci>
														<m:leq id="id91205"/>
														<m:ci id="id91206">n</m:ci>
													</m:apply>
													<m:apply id="id91208">
														<m:ci id="id91209"/>
														<m:cn id="id91210">1</m:cn>
														<m:leq id="id91212"/>
														<m:ci id="id91214">j</m:ci>
														<m:leq id="id91216"/>
														<m:ci id="id91217">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, then</p>
								<table class="equation" id="S7.Ex4">

									<tr valign="baseline" class="equation" id="S7.Ex4">
										<td id="id91237" class="eqpad"/>
										<td id="id91241" nowrap="yes" align="center" colspan="1">
											<m:math id="id91249" display="block">
												<m:semantics id="id91252">
													<m:mrow id="id91253">
														<m:mrow id="id91254">
															<m:mi id="id91255">ω</m:mi>
															<m:mo id="id91258">:=</m:mo>
															<m:mrow id="id91260">
																<m:mrow id="id91261">
																	<m:munder id="id91262">
																		<m:mo id="id91263" movablelimits="false">∑</m:mo>
																		<m:mrow id="id91268">
																			<m:mi id="id91269">i</m:mi>
																			<m:mo id="id91271">,</m:mo>
																			<m:mi id="id91273">j</m:mi>
																		</m:mrow>
																	</m:munder>
																	<m:mi id="id91275">d</m:mi>
																	<m:mo id="id91277">⁢</m:mo>
																	<m:msub id="id91280">
																		<m:mi id="id91281">x</m:mi>
																		<m:mrow id="id91283">
																			<m:mi id="id91284">i</m:mi>
																			<m:mo id="id91286">⁢</m:mo>
																			<m:mi id="id91288">j</m:mi>
																		</m:mrow>
																	</m:msub>
																</m:mrow>
																<m:mo id="id91290">∧</m:mo>
																<m:mrow id="id91293">
																	<m:mi id="id91294">d</m:mi>
																	<m:mo id="id91296">⁢</m:mo>
																	<m:msub id="id91298">
																		<m:mi id="id91300">y</m:mi>
																		<m:mrow id="id91302">
																			<m:mi id="id91303">i</m:mi>
																			<m:mo id="id91305">⁢</m:mo>
																			<m:mi id="id91307">j</m:mi>
																		</m:mrow>
																	</m:msub>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:mo id="id91309">.</m:mo>
													</m:mrow>
													<m:annotation-xml id="id91312" encoding="MathML-Content">
														<m:apply id="id91315">
															<m:ci id="id91316">:=</m:ci>
															<m:ci id="id91318">ω</m:ci>
															<m:apply id="id91320">
																<m:and id="id91322"/>
																<m:apply id="id91323">
																	<m:apply id="id91324">
																		<m:csymbol id="id91325" cd="ambiguous">subscript</m:csymbol>
																		<m:sum id="id91329"/>
																		<m:apply id="id91330">
																			<m:list id="id91332"/>
																			<m:ci id="id91333">i</m:ci>
																			<m:ci id="id91335">j</m:ci>
																		</m:apply>
																	</m:apply>
																	<m:apply id="id91337">
																		<m:times id="id91338"/>
																		<m:ci id="id91339">d</m:ci>
																		<m:apply id="id91341">
																			<m:csymbol id="id91342" cd="ambiguous">subscript</m:csymbol>
																			<m:ci id="id91347">x</m:ci>
																			<m:apply id="id91349">
																				<m:times id="id91350"/>
																				<m:ci id="id91351">i</m:ci>
																				<m:ci id="id91353">j</m:ci>
																			</m:apply>
																		</m:apply>
																	</m:apply>
																</m:apply>
																<m:apply id="id91355">
																	<m:times id="id91356"/>
																	<m:ci id="id91358">d</m:ci>
																	<m:apply id="id91360">
																		<m:csymbol id="id91361" cd="ambiguous">subscript</m:csymbol>
																		<m:ci id="id91365">y</m:ci>
																		<m:apply id="id91368">
																			<m:times id="id91369"/>
																			<m:ci id="id91370">i</m:ci>
																			<m:ci id="id91372">j</m:ci>
																		</m:apply>
																	</m:apply>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id91375" class="eqpad"/>
									</tr>

								</table>
								<p id="id91382" style="" class="p">The moment map takes values in the dual of the Lie algebra of <m:math id="id91388" display="inline">
										<m:semantics id="id91392">
											<m:mrow id="id91393">
												<m:mi id="id91394">U</m:mi>
												<m:mo id="id91396">⁢</m:mo>
												<m:mfenced id="id91398" open="(" close=")">
													<m:mi id="id91403">k</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id91406" encoding="MathML-Content">
												<m:apply id="id91409">
													<m:times id="id91410"/>
													<m:ci id="id91411">U</m:ci>
													<m:ci id="id91413">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>,
which can be identified with the space of Hermitian <m:math id="id91417" display="inline">
										<m:semantics id="id91420">
											<m:mrow id="id91421">
												<m:mi id="id91422">k</m:mi>
												<m:mo id="id91424">×</m:mo>
												<m:mi id="id91427">k</m:mi>
											</m:mrow>
											<m:annotation-xml id="id91429" encoding="MathML-Content">
												<m:apply id="id91432">
													<m:times id="id91433"/>
													<m:ci id="id91434">k</m:ci>
													<m:ci id="id91437">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> matrices.
It is a straightforward calculation using the definition of the moment
map to show that a moment map is given by</p>
								<table class="equation" id="S7.Ex5">

									<tr valign="baseline" class="equation" id="S7.Ex5">
										<td id="id91458" class="eqpad"/>
										<td id="id91462" nowrap="yes" align="center" colspan="1">
											<m:math id="id91469" display="block">
												<m:semantics id="id91473">
													<m:mrow id="id91474">
														<m:mrow id="id91475">
															<m:mrow id="id91476">
																<m:msub id="id91477">
																	<m:mi id="id91478">μ</m:mi>
																	<m:mrow id="id91480">
																		<m:mi id="id91482">U</m:mi>
																		<m:mo id="id91484">⁢</m:mo>
																		<m:mfenced id="id91486" open="(" close=")">
																			<m:mi id="id91491">k</m:mi>
																		</m:mfenced>
																	</m:mrow>
																</m:msub>
																<m:mo id="id91493">⁢</m:mo>
																<m:mfenced id="id91496" open="(" close=")">
																	<m:mi id="id91501">A</m:mi>
																</m:mfenced>
															</m:mrow>
															<m:mo id="id91503">=</m:mo>
															<m:mrow id="id91505">
																<m:mrow id="id91506">
																	<m:msup id="id91507">
																		<m:mi id="id91508">A</m:mi>
																		<m:mo id="id91510">*</m:mo>
																	</m:msup>
																	<m:mo id="id91512">⁢</m:mo>
																	<m:mi id="id91515">A</m:mi>
																</m:mrow>
																<m:mo id="id91517">-</m:mo>
																<m:mrow id="id91519">
																	<m:mn id="id91520">1</m:mn>
																	<m:mo id="id91522">⁢</m:mo>
																	<m:mi id="id91525" mathvariant="normal">l</m:mi>
																</m:mrow>
															</m:mrow>
														</m:mrow>
														<m:mo id="id91529">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id91531" encoding="MathML-Content">
														<m:apply id="id91535">
															<m:eq id="id91536"/>
															<m:apply id="id91537">
																<m:times id="id91538"/>
																<m:apply id="id91539">
																	<m:csymbol id="id91540" cd="ambiguous">subscript</m:csymbol>
																	<m:ci id="id91545">μ</m:ci>
																	<m:apply id="id91547">
																		<m:times id="id91548"/>
																		<m:ci id="id91549">U</m:ci>
																		<m:ci id="id91551">k</m:ci>
																	</m:apply>
																</m:apply>
																<m:ci id="id91553">A</m:ci>
															</m:apply>
															<m:apply id="id91556">
																<m:minus id="id91557"/>
																<m:apply id="id91558">
																	<m:times id="id91559"/>
																	<m:apply id="id91560">
																		<m:csymbol id="id91561" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id91566">A</m:ci>
																		<m:times id="id91568"/>
																	</m:apply>
																	<m:ci id="id91569">A</m:ci>
																</m:apply>
																<m:apply id="id91571">
																	<m:times id="id91572"/>
																	<m:cn id="id91573">1</m:cn>
																	<m:ci id="id91575">l</m:ci>
																</m:apply>
															</m:apply>
														</m:apply>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id91578" class="eqpad"/>
									</tr>

								</table>
								<p id="id91585" style="" class="p">where <m:math id="id91591" display="inline">
										<m:semantics id="id91595">
											<m:mrow id="id91596">
												<m:msup id="id91597">
													<m:mi id="id91598">A</m:mi>
													<m:mo id="id91600">*</m:mo>
												</m:msup>
												<m:mo id="id91602">:=</m:mo>
												<m:msup id="id91604">
													<m:mover id="id91605" accent="true">
														<m:mi id="id91609">A</m:mi>
														<m:mo id="id91611">¯</m:mo>
													</m:mover>
													<m:mo id="id91613">tr</m:mo>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id91615" encoding="MathML-Content">
												<m:apply id="id91619">
													<m:ci id="id91620">:=</m:ci>
													<m:apply id="id91622">
														<m:csymbol id="id91623" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id91628">A</m:ci>
														<m:times id="id91630"/>
													</m:apply>
													<m:apply id="id91631">
														<m:csymbol id="id91632" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id91636">
															<m:ci id="id91638">¯</m:ci>
															<m:ci id="id91640">A</m:ci>
														</m:apply>
														<m:ci id="id91642">tr</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
(precisely, given a
skew-Hermitian matrix <m:math id="id91646" display="inline">
										<m:semantics id="id91649">
											<m:mrow id="id91650">
												<m:mi id="id91651">ξ</m:mi>
												<m:mo id="id91654">∈</m:mo>
												<m:mrow id="id91656">
													<m:mo id="id91657">Lie</m:mo>
													<m:mo id="id91659">⁡</m:mo>
													<m:mfenced id="id91662" open="(" close=")">
														<m:mrow id="id91667">
															<m:mi id="id91668">U</m:mi>
															<m:mo id="id91670">⁢</m:mo>
															<m:mfenced id="id91672" open="(" close=")">
																<m:mi id="id91677">k</m:mi>
															</m:mfenced>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id91680" encoding="MathML-Content">
												<m:apply id="id91683">
													<m:ci id="id91684">∈</m:ci>
													<m:ci id="id91686">ξ</m:ci>
													<m:apply id="id91689">
														<m:ci id="id91690">Lie</m:ci>
														<m:apply id="id91692">
															<m:times id="id91693"/>
															<m:ci id="id91694">U</m:ci>
															<m:ci id="id91696">k</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, then the pairing
									<m:math id="id91700" display="inline">
										<m:semantics id="id91703">
											<m:mfenced id="id91706" open="⟨" close="⟩">
												<m:mrow id="id91712">
													<m:mrow id="id91713">
														<m:msub id="id91714">
															<m:mi id="id91715">μ</m:mi>
															<m:mrow id="id91717">
																<m:mi id="id91718">U</m:mi>
																<m:mo id="id91720">⁢</m:mo>
																<m:mfenced id="id91723" open="(" close=")">
																	<m:mi id="id91726">k</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:msub>
														<m:mo id="id91728">⁢</m:mo>
														<m:mfenced id="id91730" open="(" close=")">
															<m:mi id="id91736">A</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mo id="id91738">,</m:mo>
													<m:mi id="id91740">ξ</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id91742" encoding="MathML-Content">
												<m:apply id="id91746">
													<m:list id="id91747"/>
													<m:apply id="id91748">
														<m:times id="id91749"/>
														<m:apply id="id91750">
															<m:csymbol id="id91751" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id91756">μ</m:ci>
															<m:apply id="id91758">
																<m:times id="id91759"/>
																<m:ci id="id91760">U</m:ci>
																<m:ci id="id91762">k</m:ci>
															</m:apply>
														</m:apply>
														<m:ci id="id91764">A</m:ci>
													</m:apply>
													<m:ci id="id91766">ξ</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is given by
									<m:math id="id91770" display="inline">
										<m:semantics id="id91774">
											<m:mrow id="id91775">
												<m:mfrac id="id91776">
													<m:mi id="id91777">i</m:mi>
													<m:mn id="id91779">2</m:mn>
												</m:mfrac>
												<m:mo id="id91781">⁢</m:mo>
												<m:mrow id="id91783">
													<m:mo id="id91784">Tr</m:mo>
													<m:mo id="id91787">⁡</m:mo>
													<m:mfenced id="id91789" open="(" close=")">
														<m:mrow id="id91794">
															<m:mfenced id="id91795" open="(" close=")">
																<m:mrow id="id91800">
																	<m:mrow id="id91801">
																		<m:msup id="id91802">
																			<m:mi id="id91803">A</m:mi>
																			<m:mo id="id91806">*</m:mo>
																		</m:msup>
																		<m:mo id="id91808">⁢</m:mo>
																		<m:mi id="id91810">A</m:mi>
																	</m:mrow>
																	<m:mo id="id91812">-</m:mo>
																	<m:mrow id="id91814">
																		<m:mn id="id91815">1</m:mn>
																		<m:mo id="id91818">⁢</m:mo>
																		<m:mi id="id91820" mathvariant="normal">l</m:mi>
																	</m:mrow>
																</m:mrow>
															</m:mfenced>
															<m:mo id="id91824">⁢</m:mo>
															<m:msup id="id91827">
																<m:mi id="id91828">ξ</m:mi>
																<m:mo id="id91830">*</m:mo>
															</m:msup>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id91832" encoding="MathML-Content">
												<m:apply id="id91836">
													<m:times id="id91837"/>
													<m:apply id="id91838">
														<m:divide id="id91839"/>
														<m:ci id="id91840">i</m:ci>
														<m:cn id="id91842">2</m:cn>
													</m:apply>
													<m:apply id="id91844">
														<m:ci id="id91845">Tr</m:ci>
														<m:apply id="id91847">
															<m:times id="id91848"/>
															<m:apply id="id91850">
																<m:minus id="id91851"/>
																<m:apply id="id91852">
																	<m:times id="id91853"/>
																	<m:apply id="id91854">
																		<m:csymbol id="id91855" cd="ambiguous">superscript</m:csymbol>
																		<m:ci id="id91860">A</m:ci>
																		<m:times id="id91862"/>
																	</m:apply>
																	<m:ci id="id91863">A</m:ci>
																</m:apply>
																<m:apply id="id91865">
																	<m:times id="id91866"/>
																	<m:cn id="id91867">1</m:cn>
																	<m:ci id="id91869">l</m:ci>
																</m:apply>
															</m:apply>
															<m:apply id="id91871">
																<m:csymbol id="id91872" cd="ambiguous">superscript</m:csymbol>
																<m:ci id="id91877">ξ</m:ci>
																<m:times id="id91879"/>
															</m:apply>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>).</p>
							</div>

							<div class="para" id="S7.SS1.p2">
								<p id="id91890" style="" class="p">The <m:math id="id91897" display="inline">
										<m:semantics id="id91900">
											<m:mi id="id91901">k</m:mi>
											<m:annotation-xml id="id91903" encoding="MathML-Content">
												<m:ci id="id91907">k</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> column vectors of the matrix <m:math id="id91910" display="inline">
										<m:semantics id="id91914">
											<m:mrow id="id91915">
												<m:mi id="id91916">A</m:mi>
												<m:mo id="id91918">∈</m:mo>
												<m:mrow id="id91920">
													<m:mo id="id91921">Hom</m:mo>
													<m:mo id="id91923">⁡</m:mo>
													<m:mfenced id="id91926" open="(" close=")">
														<m:mrow id="id91931">
															<m:msup id="id91932">
																<m:mi id="id91933" mathvariant="double-struck">C</m:mi>
																<m:mi id="id91937">k</m:mi>
															</m:msup>
															<m:mo id="id91940">,</m:mo>
															<m:msup id="id91942">
																<m:mi id="id91943" mathvariant="double-struck">C</m:mi>
																<m:mi id="id91947">n</m:mi>
															</m:msup>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id91949" encoding="MathML-Content">
												<m:apply id="id91953">
													<m:ci id="id91954">∈</m:ci>
													<m:ci id="id91956">A</m:ci>
													<m:apply id="id91958">
														<m:ci id="id91959">Hom</m:ci>
														<m:apply id="id91961">
															<m:csymbol id="id91962" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id91967">C</m:ci>
															<m:ci id="id91969">k</m:ci>
														</m:apply>
														<m:apply id="id91971">
															<m:csymbol id="id91972" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id91977">C</m:ci>
															<m:ci id="id91979">n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
define vectors <m:math id="id91983" display="inline">
										<m:semantics id="id91986">
											<m:mrow id="id91987">
												<m:mrow id="id91988">
													<m:msub id="id91989">
														<m:mi id="id91990">v</m:mi>
														<m:mn id="id91992">1</m:mn>
													</m:msub>
													<m:mo id="id91995">,</m:mo>
													<m:mi id="id91997" mathvariant="normal">…</m:mi>
													<m:mo id="id92001">,</m:mo>
													<m:msub id="id92004">
														<m:mi id="id92005">v</m:mi>
														<m:mi id="id92007">k</m:mi>
													</m:msub>
												</m:mrow>
												<m:mo id="id92009">∈</m:mo>
												<m:msup id="id92011">
													<m:mi id="id92012" mathvariant="double-struck">C</m:mi>
													<m:mi id="id92017">n</m:mi>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id92019" encoding="MathML-Content">
												<m:apply id="id92022">
													<m:ci id="id92023">∈</m:ci>
													<m:apply id="id92026">
														<m:list id="id92027"/>
														<m:apply id="id92028">
															<m:csymbol id="id92029" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id92034">v</m:ci>
															<m:cn id="id92036">1</m:cn>
														</m:apply>
														<m:ci id="id92038">…</m:ci>
														<m:apply id="id92040">
															<m:csymbol id="id92041" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id92046">v</m:ci>
															<m:ci id="id92048">k</m:ci>
														</m:apply>
													</m:apply>
													<m:apply id="id92050">
														<m:csymbol id="id92051" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id92056">C</m:ci>
														<m:ci id="id92058">n</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the <m:math id="id92062" display="inline">
										<m:semantics id="id92065">
											<m:mfenced id="id92066" open="(" close=")">
												<m:mrow id="id92071">
													<m:mi id="id92072">i</m:mi>
													<m:mo id="id92074">,</m:mo>
													<m:mi id="id92076">j</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id92078" encoding="MathML-Content">
												<m:apply id="id92082">
													<m:interval id="id92083" closure="open"/>
													<m:ci id="id92086">i</m:ci>
													<m:ci id="id92088">j</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-entry of <m:math id="id92092" display="inline">
										<m:semantics id="id92095">
											<m:mrow id="id92096">
												<m:msup id="id92097">
													<m:mi id="id92098">A</m:mi>
													<m:mo id="id92100">*</m:mo>
												</m:msup>
												<m:mo id="id92103">⁢</m:mo>
												<m:mi id="id92105">A</m:mi>
											</m:mrow>
											<m:annotation-xml id="id92107" encoding="MathML-Content">
												<m:apply id="id92110">
													<m:times id="id92112"/>
													<m:apply id="id92113">
														<m:csymbol id="id92114" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id92118">A</m:ci>
														<m:times id="id92120"/>
													</m:apply>
													<m:ci id="id92122">A</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is the
Hermitian inner product <m:math id="id92125" display="inline">
										<m:semantics id="id92128">
											<m:mfenced id="id92130" open="(" close=")">
												<m:mrow id="id92135">
													<m:msub id="id92136">
														<m:mi id="id92137">v</m:mi>
														<m:mi id="id92139">j</m:mi>
													</m:msub>
													<m:mo id="id92141">,</m:mo>
													<m:msub id="id92143">
														<m:mi id="id92144">v</m:mi>
														<m:mi id="id92146">i</m:mi>
													</m:msub>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id92148" encoding="MathML-Content">
												<m:apply id="id92152">
													<m:interval id="id92153" closure="open"/>
													<m:apply id="id92156">
														<m:csymbol id="id92157" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92162">v</m:ci>
														<m:ci id="id92164">j</m:ci>
													</m:apply>
													<m:apply id="id92166">
														<m:csymbol id="id92167" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92172">v</m:ci>
														<m:ci id="id92174">i</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Hence <m:math id="id92178" display="inline">
										<m:semantics id="id92181">
											<m:mrow id="id92182">
												<m:msubsup id="id92183">
													<m:mi id="id92184">μ</m:mi>
													<m:mrow id="id92186">
														<m:mi id="id92188">U</m:mi>
														<m:mo id="id92190">⁢</m:mo>
														<m:mfenced id="id92192" open="(" close=")">
															<m:mi id="id92197">k</m:mi>
														</m:mfenced>
													</m:mrow>
													<m:mrow id="id92199">
														<m:mo id="id92200">-</m:mo>
														<m:mn id="id92202">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id92205">⁢</m:mo>
												<m:mfenced id="id92207" open="(" close=")">
													<m:mn id="id92212">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92214" encoding="MathML-Content">
												<m:apply id="id92218">
													<m:times id="id92219"/>
													<m:apply id="id92220">
														<m:csymbol id="id92221" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id92225">
															<m:csymbol id="id92226" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id92231">μ</m:ci>
															<m:apply id="id92234">
																<m:times id="id92235"/>
																<m:ci id="id92236">U</m:ci>
																<m:ci id="id92238">k</m:ci>
															</m:apply>
														</m:apply>
														<m:apply id="id92240">
															<m:minus id="id92241"/>
															<m:cn id="id92242">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id92244">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
consists of the unitary <m:math id="id92248" display="inline">
										<m:semantics id="id92251">
											<m:mi id="id92252">k</m:mi>
											<m:annotation-xml id="id92254" encoding="MathML-Content">
												<m:ci id="id92258">k</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-frames in <m:math id="id92261" display="inline">
										<m:semantics id="id92264">
											<m:msup id="id92266">
												<m:mi id="id92267" mathvariant="double-struck">C</m:mi>
												<m:mi id="id92271">n</m:mi>
											</m:msup>
											<m:annotation-xml id="id92273" encoding="MathML-Content">
												<m:apply id="id92276">
													<m:csymbol id="id92278" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id92282">C</m:ci>
													<m:ci id="id92284">n</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and taking the quotient
by <m:math id="id92288" display="inline">
										<m:semantics id="id92291">
											<m:mrow id="id92292">
												<m:mi id="id92293">U</m:mi>
												<m:mo id="id92296">⁢</m:mo>
												<m:mfenced id="id92298" open="(" close=")">
													<m:mi id="id92303">k</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92305" encoding="MathML-Content">
												<m:apply id="id92308">
													<m:times id="id92310"/>
													<m:ci id="id92311">U</m:ci>
													<m:ci id="id92313">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> gives the Grassmannian <m:math id="id92316" display="inline">
										<m:semantics id="id92320">
											<m:mrow id="id92321">
												<m:mi id="id92322">G</m:mi>
												<m:mo id="id92324">⁢</m:mo>
												<m:mfenced id="id92326" open="(" close=")">
													<m:mrow id="id92331">
														<m:mi id="id92332">k</m:mi>
														<m:mo id="id92335">,</m:mo>
														<m:mi id="id92337">n</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92339" encoding="MathML-Content">
												<m:apply id="id92342">
													<m:times id="id92343"/>
													<m:ci id="id92344">G</m:ci>
													<m:apply id="id92346">
														<m:interval id="id92348" closure="open"/>
														<m:ci id="id92351">k</m:ci>
														<m:ci id="id92353">n</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S7.SS1.p3">
								<p id="id92365" style="" class="p">The maximal torus <m:math id="id92371" display="inline">
										<m:semantics id="id92375">
											<m:mrow id="id92376">
												<m:mi id="id92377">T</m:mi>
												<m:mo id="id92379">⊂</m:mo>
												<m:mrow id="id92381">
													<m:mi id="id92382">U</m:mi>
													<m:mo id="id92385">⁢</m:mo>
													<m:mfenced id="id92387" open="(" close=")">
														<m:mi id="id92392">k</m:mi>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id92394" encoding="MathML-Content">
												<m:apply id="id92398">
													<m:ci id="id92399">⊂</m:ci>
													<m:ci id="id92401">T</m:ci>
													<m:apply id="id92403">
														<m:times id="id92404"/>
														<m:ci id="id92405">U</m:ci>
														<m:ci id="id92407">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> of diagonal matrices has associated
moment map given by the diagonal entries of the matrix <m:math id="id92412" display="inline">
										<m:semantics id="id92415">
											<m:mrow id="id92416">
												<m:mrow id="id92417">
													<m:msup id="id92418">
														<m:mi id="id92419">A</m:mi>
														<m:mo id="id92421">*</m:mo>
													</m:msup>
													<m:mo id="id92423">⁢</m:mo>
													<m:mi id="id92426">A</m:mi>
												</m:mrow>
												<m:mo id="id92428">-</m:mo>
												<m:mrow id="id92430">
													<m:mn id="id92431">1</m:mn>
													<m:mo id="id92433">⁢</m:mo>
													<m:mi id="id92436" mathvariant="normal">l</m:mi>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id92440" encoding="MathML-Content">
												<m:apply id="id92443">
													<m:minus id="id92444"/>
													<m:apply id="id92446">
														<m:times id="id92447"/>
														<m:apply id="id92448">
															<m:csymbol id="id92449" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id92453">A</m:ci>
															<m:times id="id92456"/>
														</m:apply>
														<m:ci id="id92457">A</m:ci>
													</m:apply>
													<m:apply id="id92459">
														<m:times id="id92460"/>
														<m:cn id="id92461">1</m:cn>
														<m:ci id="id92463">l</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and
so a <m:math id="id92466" display="inline">
										<m:semantics id="id92470">
											<m:mi id="id92471">k</m:mi>
											<m:annotation-xml id="id92473" encoding="MathML-Content">
												<m:ci id="id92476">k</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-tuple <m:math id="id92480" display="inline">
										<m:semantics id="id92483">
											<m:mfenced id="id92484" open="(" close=")">
												<m:mrow id="id92489">
													<m:msub id="id92490">
														<m:mi id="id92491">v</m:mi>
														<m:mn id="id92494">1</m:mn>
													</m:msub>
													<m:mo id="id92496">,</m:mo>
													<m:mi id="id92498" mathvariant="normal">…</m:mi>
													<m:mo id="id92502">,</m:mo>
													<m:msub id="id92505">
														<m:mi id="id92506">v</m:mi>
														<m:mi id="id92508">k</m:mi>
													</m:msub>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id92510" encoding="MathML-Content">
												<m:apply id="id92513">
													<m:vector id="id92514"/>
													<m:apply id="id92515">
														<m:csymbol id="id92516" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92521">v</m:ci>
														<m:cn id="id92523">1</m:cn>
													</m:apply>
													<m:ci id="id92525">…</m:ci>
													<m:apply id="id92528">
														<m:csymbol id="id92529" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92534">v</m:ci>
														<m:ci id="id92536">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> lies in <m:math id="id92539" display="inline">
										<m:semantics id="id92542">
											<m:mrow id="id92544">
												<m:msubsup id="id92545">
													<m:mi id="id92546">μ</m:mi>
													<m:mi id="id92548">T</m:mi>
													<m:mrow id="id92550">
														<m:mo id="id92551">-</m:mo>
														<m:mn id="id92553">1</m:mn>
													</m:mrow>
												</m:msubsup>
												<m:mo id="id92556">⁢</m:mo>
												<m:mfenced id="id92558" open="(" close=")">
													<m:mn id="id92563">0</m:mn>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92565" encoding="MathML-Content">
												<m:apply id="id92568">
													<m:times id="id92570"/>
													<m:apply id="id92571">
														<m:csymbol id="id92572" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id92576">
															<m:csymbol id="id92577" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id92582">μ</m:ci>
															<m:ci id="id92584">T</m:ci>
														</m:apply>
														<m:apply id="id92587">
															<m:minus id="id92588"/>
															<m:cn id="id92589">1</m:cn>
														</m:apply>
													</m:apply>
													<m:cn id="id92591">0</m:cn>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> precisely when
each <m:math id="id92594" display="inline">
										<m:semantics id="id92598">
											<m:msub id="id92599">
												<m:mi id="id92600">v</m:mi>
												<m:mi id="id92602">i</m:mi>
											</m:msub>
											<m:annotation-xml id="id92604" encoding="MathML-Content">
												<m:apply id="id92607">
													<m:csymbol id="id92608" cd="ambiguous">subscript</m:csymbol>
													<m:ci id="id92613">v</m:ci>
													<m:ci id="id92615">i</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> has length <m:math id="id92619" display="inline">
										<m:semantics id="id92622">
											<m:mn id="id92623">1</m:mn>
											<m:annotation-xml id="id92625" encoding="MathML-Content">
												<m:cn id="id92629">1</m:cn>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. The torus <m:math id="id92632" display="inline">
										<m:semantics id="id92635">
											<m:mi id="id92636">T</m:mi>
											<m:annotation-xml id="id92639" encoding="MathML-Content">
												<m:ci id="id92642">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> equals the product <m:math id="id92645" display="inline">
										<m:semantics id="id92649">
											<m:msup id="id92650">
												<m:mfenced id="id92651" open="(" close=")">
													<m:msup id="id92656">
														<m:mi id="id92657">S</m:mi>
														<m:mn id="id92659">1</m:mn>
													</m:msup>
												</m:mfenced>
												<m:mi id="id92661">k</m:mi>
											</m:msup>
											<m:annotation-xml id="id92663" encoding="MathML-Content">
												<m:apply id="id92667">
													<m:csymbol id="id92668" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id92672">
														<m:csymbol id="id92674" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id92678">S</m:ci>
														<m:cn id="id92680">1</m:cn>
													</m:apply>
													<m:ci id="id92682">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>,
the factors of which rotate the vectors independently,
hence identifying the quotient <m:math id="id92687" display="inline">
										<m:semantics id="id92690">
											<m:mrow id="id92691">
												<m:mo id="id92692">Hom</m:mo>
												<m:mo id="id92694">(</m:mo>
												<m:msup id="id92696">
													<m:mi id="id92697" mathvariant="double-struck">C</m:mi>
													<m:mi id="id92702">k</m:mi>
												</m:msup>
												<m:mo id="id92704">,</m:mo>
												<m:msup id="id92706">
													<m:mi id="id92707" mathvariant="double-struck">C</m:mi>
													<m:mi id="id92712">n</m:mi>
												</m:msup>
												<m:mo id="id92714">)</m:mo>
												<m:mo id="id92716">/</m:mo>
												<m:mo id="id92718">/</m:mo>
												<m:mi id="id92720">T</m:mi>
											</m:mrow>
											<m:annotation-xml id="id92722" encoding="MathML-Content"/>
										</m:semantics>
									</m:math>
with <m:math id="id92727" display="inline">
										<m:semantics id="id92730">
											<m:msup id="id92731">
												<m:mfenced id="id92732" open="(" close=")">
													<m:mrow id="id92737">
														<m:mi id="id92738" mathvariant="double-struck">C</m:mi>
														<m:mo id="id92743">⁢</m:mo>
														<m:msup id="id92745">
															<m:mi id="id92746" mathvariant="double-struck">P</m:mi>
															<m:mrow id="id92751">
																<m:mi id="id92752">n</m:mi>
																<m:mo id="id92754">-</m:mo>
																<m:mn id="id92756">1</m:mn>
															</m:mrow>
														</m:msup>
													</m:mrow>
												</m:mfenced>
												<m:mi id="id92758">k</m:mi>
											</m:msup>
											<m:annotation-xml id="id92760" encoding="MathML-Content">
												<m:apply id="id92764">
													<m:csymbol id="id92765" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id92769">
														<m:times id="id92770"/>
														<m:ci id="id92772">C</m:ci>
														<m:apply id="id92774">
															<m:csymbol id="id92775" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id92779">P</m:ci>
															<m:apply id="id92782">
																<m:minus id="id92783"/>
																<m:ci id="id92784">n</m:ci>
																<m:cn id="id92786">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:ci id="id92788">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. To calculate the classes <m:math id="id92792" display="inline">
										<m:semantics id="id92795">
											<m:mrow id="id92796">
												<m:mi id="id92797">e</m:mi>
												<m:mo id="id92799">⁢</m:mo>
												<m:mfenced id="id92802" open="(" close=")">
													<m:mi id="id92807">α</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92809" encoding="MathML-Content">
												<m:apply id="id92812">
													<m:times id="id92813"/>
													<m:ci id="id92814">e</m:ci>
													<m:ci id="id92817">α</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, consider the
conjugation action of the diagonal matrices on the skew-symmetric
matrices whose diagonal entries vanish (this is the complement of <m:math id="id92821" display="inline">
										<m:semantics id="id92825">
											<m:mi id="id92826" mathvariant="fraktur">t</m:mi>
											<m:annotation-xml id="id92830" encoding="MathML-Content">
												<m:ci id="id92834">t</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
in <m:math id="id92837" display="inline">
										<m:semantics id="id92840">
											<m:mrow id="id92841">
												<m:mo id="id92842">Lie</m:mo>
												<m:mo id="id92845">⁡</m:mo>
												<m:mfenced id="id92847" open="(" close=")">
													<m:mrow id="id92852">
														<m:mi id="id92853">U</m:mi>
														<m:mo id="id92855">⁢</m:mo>
														<m:mfenced id="id92858" open="(" close=")">
															<m:mi id="id92863">k</m:mi>
														</m:mfenced>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id92865" encoding="MathML-Content">
												<m:apply id="id92868">
													<m:ci id="id92869">Lie</m:ci>
													<m:apply id="id92871">
														<m:times id="id92872"/>
														<m:ci id="id92874">U</m:ci>
														<m:ci id="id92876">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. The matrix with diagonal entries <m:math id="id92879" display="inline">
										<m:semantics id="id92883">
											<m:mfenced id="id92884" open="(" close=")">
												<m:mrow id="id92889">
													<m:msub id="id92890">
														<m:mi id="id92891">λ</m:mi>
														<m:mn id="id92893">1</m:mn>
													</m:msub>
													<m:mo id="id92895">,</m:mo>
													<m:mi id="id92898" mathvariant="normal">…</m:mi>
													<m:mo id="id92902">,</m:mo>
													<m:msub id="id92904">
														<m:mi id="id92905">λ</m:mi>
														<m:mi id="id92908">k</m:mi>
													</m:msub>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id92910" encoding="MathML-Content">
												<m:apply id="id92913">
													<m:vector id="id92914"/>
													<m:apply id="id92915">
														<m:csymbol id="id92916" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92921">λ</m:ci>
														<m:cn id="id92924">1</m:cn>
													</m:apply>
													<m:ci id="id92926">…</m:ci>
													<m:apply id="id92928">
														<m:csymbol id="id92929" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id92934">λ</m:ci>
														<m:ci id="id92936">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>
acts by on the <m:math id="id92940" display="inline">
										<m:semantics id="id92943">
											<m:mfenced id="id92944" open="(" close=")">
												<m:mrow id="id92949">
													<m:mi id="id92950">i</m:mi>
													<m:mo id="id92952">,</m:mo>
													<m:mi id="id92954">j</m:mi>
												</m:mrow>
											</m:mfenced>
											<m:annotation-xml id="id92957" encoding="MathML-Content">
												<m:apply id="id92960">
													<m:interval id="id92961" closure="open"/>
													<m:ci id="id92964">i</m:ci>
													<m:ci id="id92966">j</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-entry by <m:math id="id92970" display="inline">
										<m:semantics id="id92973">
											<m:mrow id="id92974">
												<m:msub id="id92975">
													<m:mi id="id92976">λ</m:mi>
													<m:mi id="id92979">i</m:mi>
												</m:msub>
												<m:mo id="id92981">⁢</m:mo>
												<m:msubsup id="id92983">
													<m:mi id="id92984">λ</m:mi>
													<m:mi id="id92987">j</m:mi>
													<m:mrow id="id92989">
														<m:mo id="id92990">-</m:mo>
														<m:mn id="id92992">1</m:mn>
													</m:mrow>
												</m:msubsup>
											</m:mrow>
											<m:annotation-xml id="id92994" encoding="MathML-Content">
												<m:apply id="id92998">
													<m:times id="id92999"/>
													<m:apply id="id93000">
														<m:csymbol id="id93001" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id93006">λ</m:ci>
														<m:ci id="id93008">i</m:ci>
													</m:apply>
													<m:apply id="id93010">
														<m:csymbol id="id93011" cd="ambiguous">superscript</m:csymbol>
														<m:apply id="id93016">
															<m:csymbol id="id93017" cd="ambiguous">subscript</m:csymbol>
															<m:ci id="id93022">λ</m:ci>
															<m:ci id="id93024">j</m:ci>
														</m:apply>
														<m:apply id="id93026">
															<m:minus id="id93027"/>
															<m:cn id="id93028">1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>; the complex line
bundle constructed with this weight has Euler class <m:math id="id93032" display="inline">
										<m:semantics id="id93035">
											<m:mrow id="id93036">
												<m:msub id="id93038">
													<m:mi id="id93039">u</m:mi>
													<m:mi id="id93041">j</m:mi>
												</m:msub>
												<m:mo id="id93043">-</m:mo>
												<m:msub id="id93045">
													<m:mi id="id93046">u</m:mi>
													<m:mi id="id93048">i</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id93050" encoding="MathML-Content">
												<m:apply id="id93054">
													<m:minus id="id93055"/>
													<m:apply id="id93056">
														<m:csymbol id="id93057" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id93062">u</m:ci>
														<m:ci id="id93064">j</m:ci>
													</m:apply>
													<m:apply id="id93066">
														<m:csymbol id="id93067" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id93072">u</m:ci>
														<m:ci id="id93074">i</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.</p>
							</div>

							<div class="para" id="S7.SS1.p4">
								<p id="id93086" style="" class="p">We now describe the tautological bundle <m:math id="id93092" display="inline">
										<m:semantics id="id93096">
											<m:mrow id="id93097">
												<m:mi id="id93098">E</m:mi>
												<m:mo id="id93100">→</m:mo>
												<m:mrow id="id93102">
													<m:mi id="id93103">G</m:mi>
													<m:mo id="id93106">⁢</m:mo>
													<m:mfenced id="id93108" open="(" close=")">
														<m:mrow id="id93113">
															<m:mi id="id93114">k</m:mi>
															<m:mo id="id93116">,</m:mo>
															<m:mi id="id93118">n</m:mi>
														</m:mrow>
													</m:mfenced>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id93120" encoding="MathML-Content">
												<m:apply id="id93124">
													<m:ci id="id93125">→</m:ci>
													<m:ci id="id93127">E</m:ci>
													<m:apply id="id93129">
														<m:times id="id93130"/>
														<m:ci id="id93132">G</m:ci>
														<m:apply id="id93134">
															<m:interval id="id93135" closure="open"/>
															<m:ci id="id93138">k</m:ci>
															<m:ci id="id93140">n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in terms of the
symplectic quotient construction, so that we can identify the Chern
classes of its dual on the <m:math id="id93144" display="inline">
										<m:semantics id="id93148">
											<m:mi id="id93149">T</m:mi>
											<m:annotation-xml id="id93151" encoding="MathML-Content">
												<m:ci id="id93154">T</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-symplectic quotient <m:math id="id93158" display="inline">
										<m:semantics id="id93161">
											<m:msup id="id93162">
												<m:mfenced id="id93163" open="(" close=")">
													<m:mrow id="id93168">
														<m:mi id="id93169" mathvariant="double-struck">C</m:mi>
														<m:mo id="id93174">⁢</m:mo>
														<m:msup id="id93176">
															<m:mi id="id93177" mathvariant="double-struck">P</m:mi>
															<m:mrow id="id93182">
																<m:mi id="id93183">n</m:mi>
																<m:mo id="id93185">-</m:mo>
																<m:mn id="id93187">1</m:mn>
															</m:mrow>
														</m:msup>
													</m:mrow>
												</m:mfenced>
												<m:mi id="id93189">k</m:mi>
											</m:msup>
											<m:annotation-xml id="id93191" encoding="MathML-Content">
												<m:apply id="id93195">
													<m:csymbol id="id93196" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id93200">
														<m:times id="id93201"/>
														<m:ci id="id93202">C</m:ci>
														<m:apply id="id93205">
															<m:csymbol id="id93206" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id93210">P</m:ci>
															<m:apply id="id93212">
																<m:minus id="id93214"/>
																<m:ci id="id93215">n</m:ci>
																<m:cn id="id93217">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:ci id="id93219">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
A point of <m:math id="id93222" display="inline">
										<m:semantics id="id93226">
											<m:mrow id="id93227">
												<m:mi id="id93228">G</m:mi>
												<m:mo id="id93230">⁢</m:mo>
												<m:mfenced id="id93232" open="(" close=")">
													<m:mrow id="id93237">
														<m:mi id="id93238">k</m:mi>
														<m:mo id="id93241">,</m:mo>
														<m:mi id="id93243">n</m:mi>
													</m:mrow>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id93245" encoding="MathML-Content">
												<m:apply id="id93248">
													<m:times id="id93249"/>
													<m:ci id="id93250">G</m:ci>
													<m:apply id="id93252">
														<m:interval id="id93254" closure="open"/>
														<m:ci id="id93257">k</m:ci>
														<m:ci id="id93259">n</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is a <m:math id="id93262" display="inline">
										<m:semantics id="id93266">
											<m:mrow id="id93267">
												<m:mi id="id93268">U</m:mi>
												<m:mo id="id93270">⁢</m:mo>
												<m:mfenced id="id93272" open="(" close=")">
													<m:mi id="id93278">k</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id93280" encoding="MathML-Content">
												<m:apply id="id93283">
													<m:times id="id93284"/>
													<m:ci id="id93285">U</m:ci>
													<m:ci id="id93287">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-orbit of nondegenerate homomorphisms
									<m:math id="id93291" display="inline">
										<m:semantics id="id93294">
											<m:mrow id="id93295">
												<m:mi id="id93296">A</m:mi>
												<m:mo id="id93298">:</m:mo>
												<m:mrow id="id93301">
													<m:msup id="id93302">
														<m:mi id="id93303" mathvariant="double-struck">C</m:mi>
														<m:mi id="id93307">k</m:mi>
													</m:msup>
													<m:mo id="id93309">→</m:mo>
													<m:msup id="id93312">
														<m:mi id="id93313" mathvariant="double-struck">C</m:mi>
														<m:mi id="id93317">n</m:mi>
													</m:msup>
												</m:mrow>
											</m:mrow>
											<m:annotation-xml id="id93319" encoding="MathML-Content">
												<m:apply id="id93323">
													<m:ci id="id93324">:</m:ci>
													<m:ci id="id93326">A</m:ci>
													<m:apply id="id93328">
														<m:ci id="id93329">→</m:ci>
														<m:apply id="id93331">
															<m:csymbol id="id93332" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id93337">C</m:ci>
															<m:ci id="id93339">k</m:ci>
														</m:apply>
														<m:apply id="id93341">
															<m:csymbol id="id93342" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id93347">C</m:ci>
															<m:ci id="id93349">n</m:ci>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and the corresponding fibre of <m:math id="id93353" display="inline">
										<m:semantics id="id93356">
											<m:mi id="id93357">E</m:mi>
											<m:annotation-xml id="id93359" encoding="MathML-Content">
												<m:ci id="id93363">E</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is
									<m:math id="id93366" display="inline">
										<m:semantics id="id93370">
											<m:mrow id="id93371">
												<m:mrow id="id93372">
													<m:mo id="id93373">im</m:mo>
													<m:mo id="id93375">⁡</m:mo>
													<m:mfenced id="id93377" open="(" close=")">
														<m:mi id="id93382">A</m:mi>
													</m:mfenced>
												</m:mrow>
												<m:mo id="id93384">⊂</m:mo>
												<m:msup id="id93387">
													<m:mi id="id93388" mathvariant="double-struck">C</m:mi>
													<m:mi id="id93392">n</m:mi>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id93394" encoding="MathML-Content">
												<m:apply id="id93398">
													<m:ci id="id93399">⊂</m:ci>
													<m:apply id="id93401">
														<m:ci id="id93402">im</m:ci>
														<m:ci id="id93404">A</m:ci>
													</m:apply>
													<m:apply id="id93407">
														<m:csymbol id="id93408" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id93412">C</m:ci>
														<m:ci id="id93414">n</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Two points <m:math id="id93418" display="inline">
										<m:semantics id="id93421">
											<m:mi id="id93422">A</m:mi>
											<m:annotation-xml id="id93424" encoding="MathML-Content">
												<m:ci id="id93428">A</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> and <m:math id="id93431" display="inline">
										<m:semantics id="id93435">
											<m:mrow id="id93436">
												<m:mi id="id93437">A</m:mi>
												<m:mo id="id93439">⁢</m:mo>
												<m:msup id="id93441">
													<m:mi id="id93442">g</m:mi>
													<m:mrow id="id93444">
														<m:mo id="id93446">-</m:mo>
														<m:mn id="id93448">1</m:mn>
													</m:mrow>
												</m:msup>
											</m:mrow>
											<m:annotation-xml id="id93450" encoding="MathML-Content">
												<m:apply id="id93453">
													<m:times id="id93454"/>
													<m:ci id="id93455">A</m:ci>
													<m:apply id="id93457">
														<m:csymbol id="id93458" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id93463">g</m:ci>
														<m:apply id="id93465">
															<m:minus id="id93466"/>
															<m:cn id="id93467">1</m:cn>
														</m:apply>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> in the same
									<m:math id="id93471" display="inline">
										<m:semantics id="id93474">
											<m:mrow id="id93475">
												<m:mi id="id93476">U</m:mi>
												<m:mo id="id93478">⁢</m:mo>
												<m:mfenced id="id93481" open="(" close=")">
													<m:mi id="id93486">k</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id93488" encoding="MathML-Content">
												<m:apply id="id93491">
													<m:times id="id93492"/>
													<m:ci id="id93494">U</m:ci>
													<m:ci id="id93496">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>-orbit give different identifications of <m:math id="id93499" display="inline">
										<m:semantics id="id93503">
											<m:msup id="id93504">
												<m:mi id="id93505" mathvariant="double-struck">C</m:mi>
												<m:mi id="id93509">k</m:mi>
											</m:msup>
											<m:annotation-xml id="id93511" encoding="MathML-Content">
												<m:apply id="id93515">
													<m:csymbol id="id93516" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id93520">C</m:ci>
													<m:ci id="id93523">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> with the
subspace <m:math id="id93526" display="inline">
										<m:semantics id="id93529">
											<m:mrow id="id93530">
												<m:mo id="id93532">im</m:mo>
												<m:mo id="id93534">⁡</m:mo>
												<m:mfenced id="id93536" open="(" close=")">
													<m:mi id="id93541">A</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id93543" encoding="MathML-Content">
												<m:apply id="id93547">
													<m:ci id="id93548">im</m:ci>
													<m:ci id="id93550">A</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>, and, taking this into account gives</p>
								<table class="equation" id="S7.Ex6">

									<tr valign="baseline" class="equation" id="S7.Ex6">
										<td id="id93570" class="eqpad"/>
										<td id="id93574" nowrap="yes" align="center" colspan="1">
											<m:math id="id93582" display="block">
												<m:semantics id="id93585">
													<m:mrow id="id93586">
														<m:mi id="id93587">E</m:mi>
														<m:mo id="id93589">≅</m:mo>
														<m:msup id="id93592">
															<m:mi id="id93593">μ</m:mi>
															<m:mrow id="id93595">
																<m:mo id="id93596">-</m:mo>
																<m:mn id="id93598">1</m:mn>
															</m:mrow>
														</m:msup>
														<m:mo id="id93601">(</m:mo>
														<m:mn id="id93603">0</m:mn>
														<m:mo id="id93605">)</m:mo>
														<m:msub id="id93607">
															<m:mo id="id93608">×</m:mo>
															<m:mrow id="id93610">
																<m:mi id="id93612">U</m:mi>
																<m:mo id="id93614">⁢</m:mo>
																<m:mfenced id="id93616" open="(" close=")">
																	<m:mi id="id93621">k</m:mi>
																</m:mfenced>
															</m:mrow>
														</m:msub>
														<m:msubsup id="id93623">
															<m:mi id="id93624" mathvariant="double-struck">C</m:mi>
															<m:mfenced id="id93629" open="(" close=")">
																<m:mtext id="id93634">def.</m:mtext>
															</m:mfenced>
															<m:mi id="id93636">k</m:mi>
														</m:msubsup>
														<m:mo id="id93638">,</m:mo>
													</m:mrow>
													<m:annotation-xml id="id93640" encoding="MathML-Content">
														<m:ci id="id93644">E</m:ci>
													</m:annotation-xml>
												</m:semantics>
											</m:math>
										</td>
										<td id="id93647" class="eqpad"/>
									</tr>

								</table>
								<p id="id93654" style="" class="p">where <m:math id="id93660" display="inline">
										<m:semantics id="id93663">
											<m:msubsup id="id93664">
												<m:mi id="id93665" mathvariant="double-struck">C</m:mi>
												<m:mfenced id="id93670" open="(" close=")">
													<m:mtext id="id93675">def.</m:mtext>
												</m:mfenced>
												<m:mi id="id93677">k</m:mi>
											</m:msubsup>
											<m:annotation-xml id="id93679" encoding="MathML-Content">
												<m:apply id="id93683">
													<m:csymbol id="id93684" cd="ambiguous">subscript</m:csymbol>
													<m:apply id="id93688">
														<m:csymbol id="id93690" cd="ambiguous">superscript</m:csymbol>
														<m:ci id="id93694">C</m:ci>
														<m:ci id="id93696">k</m:ci>
													</m:apply>
													<m:mtext id="id93698">def.</m:mtext>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> denotes the defining representation of
									<m:math id="id93702" display="inline">
										<m:semantics id="id93706">
											<m:mrow id="id93707">
												<m:mi id="id93708">U</m:mi>
												<m:mo id="id93710">⁢</m:mo>
												<m:mfenced id="id93712" open="(" close=")">
													<m:mi id="id93718">k</m:mi>
												</m:mfenced>
											</m:mrow>
											<m:annotation-xml id="id93720" encoding="MathML-Content">
												<m:apply id="id93723">
													<m:times id="id93724"/>
													<m:ci id="id93725">U</m:ci>
													<m:ci id="id93727">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>. Thus <m:math id="id93731" display="inline">
										<m:semantics id="id93734">
											<m:msup id="id93735">
												<m:mi id="id93736">E</m:mi>
												<m:mo id="id93738">*</m:mo>
											</m:msup>
											<m:annotation-xml id="id93740" encoding="MathML-Content">
												<m:apply id="id93744">
													<m:csymbol id="id93745" cd="ambiguous">superscript</m:csymbol>
													<m:ci id="id93750">E</m:ci>
													<m:times id="id93752"/>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> is constructed from the  dual of the defining
representation, and when we restrict this dual representation to the
maximal torus, it decomposes into <m:math id="id93755" display="inline">
										<m:semantics id="id93758">
											<m:mi id="id93760">k</m:mi>
											<m:annotation-xml id="id93762" encoding="MathML-Content">
												<m:ci id="id93765">k</m:ci>
											</m:annotation-xml>
										</m:semantics>
									</m:math> one-dimensional representations,
which have associated line bundles on the
									<m:math id="id93769" display="inline">
										<m:semantics id="id93772">
											<m:msup id="id93773">
												<m:mfenced id="id93774" open="(" close=")">
													<m:mrow id="id93780">
														<m:mi id="id93781" mathvariant="double-struck">C</m:mi>
														<m:mo id="id93785">⁢</m:mo>
														<m:msup id="id93787">
															<m:mi id="id93788" mathvariant="double-struck">P</m:mi>
															<m:mrow id="id93793">
																<m:mi id="id93794">n</m:mi>
																<m:mo id="id93796">-</m:mo>
																<m:mn id="id93798">1</m:mn>
															</m:mrow>
														</m:msup>
													</m:mrow>
												</m:mfenced>
												<m:mi id="id93800">k</m:mi>
											</m:msup>
											<m:annotation-xml id="id93802" encoding="MathML-Content">
												<m:apply id="id93806">
													<m:csymbol id="id93807" cd="ambiguous">superscript</m:csymbol>
													<m:apply id="id93812">
														<m:times id="id93813"/>
														<m:ci id="id93814">C</m:ci>
														<m:apply id="id93816">
															<m:csymbol id="id93817" cd="ambiguous">superscript</m:csymbol>
															<m:ci id="id93822">P</m:ci>
															<m:apply id="id93824">
																<m:minus id="id93825"/>
																<m:ci id="id93826">n</m:ci>
																<m:cn id="id93828">1</m:cn>
															</m:apply>
														</m:apply>
													</m:apply>
													<m:ci id="id93830">k</m:ci>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math> with Euler classes <m:math id="id93834" display="inline">
										<m:semantics id="id93837">
											<m:mrow id="id93838">
												<m:msub id="id93839">
													<m:mi id="id93840">u</m:mi>
													<m:mn id="id93842">1</m:mn>
												</m:msub>
												<m:mo id="id93844">,</m:mo>
												<m:mi id="id93846" mathvariant="normal">…</m:mi>
												<m:mo id="id93851">,</m:mo>
												<m:msub id="id93853">
													<m:mi id="id93854">u</m:mi>
													<m:mi id="id93856">k</m:mi>
												</m:msub>
											</m:mrow>
											<m:annotation-xml id="id93859" encoding="MathML-Content">
												<m:apply id="id93862">
													<m:list id="id93863"/>
													<m:apply id="id93864">
														<m:csymbol id="id93865" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id93870">u</m:ci>
														<m:cn id="id93872">1</m:cn>
													</m:apply>
													<m:ci id="id93874">…</m:ci>
													<m:apply id="id93876">
														<m:csymbol id="id93878" cd="ambiguous">subscript</m:csymbol>
														<m:ci id="id93882">u</m:ci>
														<m:ci id="id93884">k</m:ci>
													</m:apply>
												</m:apply>
											</m:annotation-xml>
										</m:semantics>
									</m:math>.
The identification of the Chern classes as elementary symmetric
polynomials then follows.</p>
							</div>

						</div>

					</div>

					<div class="bibliography" id="bib">
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